The trigonometric formulas for ratios are majorly based on the three sides of a right-angled triangle, such as the adjacent side or base, perpendicular and hypotenuse (See the above figure). Applying Pythagoras theorem for the given right-angled triangle, we have: (Perpendicular) 2 + (Base) 2 = (Hypotenuse) 2. ⇒ (P) 2 + (B) 2 = (H) 2.Trigonometry Ratios-Sine, Cosine, Tangent. The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle.Discover how to measure angles, distances, and heights using trigonometric ratios and the unit circle. Learn how to use sine, cosine, and tangent to solve real-world problems involving triangles and circular motion.Trigonometry is a branch of mathematics that deals with the relationship between sides and angles connected through ratios. It moreover helps in the calculation of angles and sides of a triangle with the help of different trigonometric ratios.It is applied in different domains including those of engineering, architecture, physics, surveying, …Once you know the value of sine and cosine, you can use the following trigonometric identities to obtain the values of the other four functions: Tangent is the sine-to-cosine ratio. tan(α) = sin(α)/cos(α) Cosecant is the reciprocal of the sine. csc(α) = 1/sin(α) Secant is the reciprocal of the cosine. sec(α) = 1/cos(α)5 Jan 2022 ... Trigonometric Ratios - Math's Working Model - TLM project | howtofunda @craftpiller #trignometricratios #mathsworkingmodel #tlmproject ...Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. Here the basic trigonometric function of Sin θ = x, can be changed to Sin-1 x = θ. Here x can have values in whole numbers, decimals, fractions, or exponents. For θ = 30° we have θ = Sin-1 (1/2). All the trigonometric formulas can be transformed into ...Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles ...List of Important Trigonometry Formulas. First let us learn basic formulas of trigonometry, considering a right-angled triangle, which has an angle θ, a hypotenuse, a side opposite angle to angle θ and a side adjacent to angle θ. Trigonometric Ratios. So the general trigonometry ratios for a right-angled triangle can be written as;The hypotenuse is NM. The hypotenuse is LN. The side adjacent ∠L is NM. The side adjacent ∠N is ML., Identify the triangle that contains an acute angle for which the sine and cosine ratios are equal., What are the values of the three trigonometric ratios for angle L, in simplest form? sin(L) = cos(L) = tan(L) = and more.Inverse trigonometric ratios are the inverse of the trigonometric functions operating on the ratio of the sides of the triangle to find out the measure of the angles of the right-angled triangle. The inverse of a function is denoted by the superscript "-1" of the given trigonometric function. For example, the inverse of the cosine function will be cos-1.In a right triangle, the hypotenuse is the longest side, an "opposite" side is the one across from a given angle, and an "adjacent" side is next to a given angle. We use special words to describe the sides of right triangles. The hypotenuse of a right triangle is always the side opposite the right angle. It is the longest side in a right triangle.14 Feb 2024 ... Trigonometric Ratios. No views · 12 minutes ago ...more. Daniel Judge ... Basic trigonometry | Basic trigonometry | Trigonometry | Khan Academy.Trigonometry is the branch of mathematics that deals with the study of the relationship between the sides of a triangle (right-angled triangle) and its angles. The relationship is presented as the ratio of the sides, which are trigonometric ratios. The six trigonometric ratios are sine, cosine, tangent, cotangent, secant, and cosecant.Accurate trigonometric ratios for 0°, 30°, 45°, 60° and 90° The trigonometric ratios for the angles 30°, 45° and 60° can be calculated using two special triangles. Similar Triangles and Trigonometric Ratios. A right triangle is a triangle with one of the angles being 90o . This is a special angle and the side opposite is called the hypotenuse. There are two other angles which are not distinguished but we may name them or assign a variable to represent their measure.If you feel like your to-do list is full of items that simply fulfill tasks other people have hoisted on you and not enough items that fulfill your own goals, writer and entreprene...Mar 4, 2023 · 1 Use measurements to calculate the trigonometric ratios for acute angles #1-10, 57-60. 2 Use trigonometric ratios to find unknown sides of right triangles #11-26. 3 Solve problems using trigonometric ratios #27-34, 41-46. 4 Use trig ratios to write equations relating the sides of a right triangle #35-40. 14 Apr 2023 ... Trigonometry Ratios Formula · Trigonometrical Functions of angle. · Relation b/w hypotenuse , base and perpendicular. · T – Tatio Formula.2.9: Trigonometric Functions. Identify the hypotenuse, adjacent side, and opposite side of an acute angle in a right triangle. Determine the three trigonometric ratios for a given angle in a right triangle. Use a calculator to find the value of the three trigonometric functions for any acute angle.The trigonometric ratios sin, cos and tan are defined as follows: Sine is the ratio between the Opposite side and the Hypothenuse, Cosine is the ratio between the Adjacent side and the Hypothenuse ...1 Use measurements to calculate the trigonometric ratios for acute angles #1-10, 57-60. 2 Use trigonometric ratios to find unknown sides of right triangles #11-26. 3 Solve problems using trigonometric ratios #27-34, 41-46. 4 Use trig ratios to write equations relating the sides of a right triangle #35-40.Trigonometry Ratios-Sine, Cosine, Tangent. The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle. If you’re shopping for a new mortgage, you may have heard of the debt-to-income ratio. So, what is it and why does it affect your mortgage? We have all your questions answered. You...About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright ...Usually, an ROA ratio, or return on assets ratio, is considered “good” if it is above five percent. An ROA ratio is a measure of how much profit a company generated for each dollar...Trigonometric ratios table helps to find the values of trigonometric standard angles such as 0°, 30°, 45°, 60° and 90°. It consists of trigonometric ratios – sine, cosine, tangent, cosecant, secant, cotangent. These ratios can be written in short as sin, cos, tan, cosec, sec and cot. The values of trigonometric ratios of standard angles ...William Sharpe's Sharpe ratio is a measure of risk-adjusted returns used to determine the best or worst returns given volatility within a market. The Sharpe ratio measures the abil...But the value of this term is also 1. For this, please go through the trigonometric ratios of specific angles. Thus, E = 1 E = 1. Challenge 2: Find the value of the following expression: E = …Nov 21, 2023 · The trigonometric ratios sin, cos and tan are defined as follows: Sine is the ratio between the Opposite side and the Hypothenuse, Cosine is the ratio between the Adjacent side and the Hypothenuse ... The trigonometric formulas for ratios are majorly based on the three sides of a right-angled triangle, such as the adjacent side or base, perpendicular and hypotenuse (See the above figure). Applying Pythagoras theorem for the given right-angled triangle, we have: (Perpendicular) 2 + (Base) 2 = (Hypotenuse) 2. ⇒ (P) 2 + (B) 2 = (H) 2.Comparing line items in your financial statements can help you make better business decisions. Learn what financial ratio analysis is. Accounting | What is Download our FREE Guide ...The trigonometric ratios can be used to solve 3-dimensional close three-dimensional (3D) An object with width, height and depth, eg a cube. problems which involve calculating a length or an angle ...This math video tutorial provides a basic introduction into trigonometry. It covers trigonometric ratios such as sine, cosine, and tangent. It explains how...The trigonometric formulas for ratios are majorly based on the three sides of a right-angled triangle, such as the adjacent side or base, perpendicular and hypotenuse (See the above figure). Applying Pythagoras theorem for the given right-angled triangle, we have: (Perpendicular) 2 + (Base) 2 = (Hypotenuse) 2. ⇒ (P) 2 + (B) 2 = (H) 2.A look at what's happened in the market six times when the equity put/call ratio has been lower than 40%. One market indicator I frequently look at on Real Money is the equity ...14 Apr 2023 ... Trigonometry Ratios Formula · Trigonometrical Functions of angle. · Relation b/w hypotenuse , base and perpendicular. · T – Tatio Formula.1. Choose which trig ratio to use. - Choose sin, cos, or tan 2. Substitute - Write the trig ratio and substitute in the values 3. Solve - Solve for the angle using the inverse ratios. The inverse ratios start with the ratio and then find the angle that produces this ratio. They are sometimes also called the arcsin, arccos, and arctan.The three basic trigonometric ratios are called sine , cosine, and tangent. Each of these represents a proportion between the lengths of two of a triangle's sides as shown below: Sine is the length of the leg opposite to the angle divided by the length of the hypotenuse. Abbreviated "sin." Trigonometric Ratios of Complementary Angles Examples. To have a better insight on trigonometric ratios of complementary angles consider the following example. Example: If A, B and C are the interior angles of a right-angle triangle, right-angled at B then find the value of A, given that tan 2A = cot(A – 30°) and 2A is an acute angle. ...Once you know the value of sine and cosine, you can use the following trigonometric identities to obtain the values of the other four functions: Tangent is the sine-to-cosine ratio. tan(α) = sin(α)/cos(α) Cosecant is the reciprocal of the sine. csc(α) = 1/sin(α) Secant is the reciprocal of the cosine. sec(α) = 1/cos(α)Meetings where decisions are made but no one wants to be accountable for them, are the worst meetings. Keep that “action to decision” ratio low by ending every meeting with an “acc...Trigonometric Values of Standard Angles. The following table gives the values of trigonometric ratios for standard angles. Degrees Radians Sine Cosine Tangent 0° 0 0 1 0 30° π 6 12 3√ 2 3√ 3 45° π 4 2√ 2 2√ 2 1 60° π 3 3√ 2 12 3√ 90° π 2 1 0 undefined 180° π 0 −1 0 270° 3π 2 −1 0 undefined 360° 2π 0 1 0 Degrees 0 ...The trigonometric ratios only work for the non-right angles, in this case either angle 𝐴 or angle 𝐵. We know that the measure of angle 𝐵 is 50°, so let's use that. – – – In a triangle, the side opposite of an angle is the side that does not help form the angle.You see gears in just about anything that has spinning parts — car engines, transmissions, electric toothbrushes... Learn what these gears are doing and get to the bottom of the "g...Find the values of the other four trigonometric ratios. So for the trig ratios, I like to use SOH-CAH-TOA to remember what the definitions of the trig ratios were So let me write this down, SOH, CAH I'll write CAH in a different color, SOH, CAH that's not a different color I'm trying- I'm having trouble changing colors! SOH, CAH, TOA. SOH, CAH ...Aug 3, 2023 · Trigonometric ratios are the ratios of the side lengths of a right-angled triangle. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). There are three more trigonometric ratios, cosecant (cosec), secant (sec), and cotangent (cot), that is, the inverse of sine, cosine, and tangent. Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical ... Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles ...A qualification ratio is actually two ratios that banks use to determine whether a borrower is eligible for a mortgage. A qualification ratio is actually two ratios that banks use ...Trigonometry Ratios-Sine, Cosine, Tangent. The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle. Trigonometric functions include trigonometric ratios, compound angles and multiple angles, which are used to get consistent results. This article explains Trigonometric Ratios of Compound Angles along with solved examples. An angle is a ray rotated about the initial point. The point where the rotation starts is the initial side, and the point ...17 Mar 2022 ... Deriving the special angle trigonometric ratios by constructing the 30°-60°-90° and 45°-45°-90° triangles. Link to Trigonometry playlist ...Trigonometry Ratios-Sine, Cosine, Tangent. The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle. Find trigonometric ratios using the Pythagorean theorem. tan ( θ) = 7 24 for an acute angle θ . Find sec ( θ) without the use of a calculator. Stuck? Use a hint. Loading... Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the ... 8.2: The Trigonometric Ratios. There are six common trigonometric ratios that relate the sides of a right triangle to the angles within the triangle. The three standard ratios are the sine, cosine and tangent. These are often abbreviated sin, cos and tan. For example, 120° and 30° are allied angles because their difference is 90°. Using the above points, we can say six trigonometric ratios sine, cosine, tangent, cotangent, cosecant, and secant. And when the two angles represent either 0°, 90°, or multiple of 90° in their addition or subtraction can be called allied angles.17 Mar 2022 ... Deriving the special angle trigonometric ratios by constructing the 30°-60°-90° and 45°-45°-90° triangles. Link to Trigonometry playlist ...This math video tutorial provides a basic introduction into trigonometry. It covers trigonometric ratios such as sine, cosine, and tangent. It explains how...The six basic trigonometric functions are: 1. Sine, #sintheta# 2. Cosine, #costheta# 3. Tangent, #tantheta# 4. Cotangent, #cottheta# 5. Secant, #sectheta# 6. Cosecant, #csctheta# Take the following triangle for example: Let the angle marked at A be #theta#. The longest side of the triangle is the hypotenuse, the side next to the angle is the …29 Oct 2017 ... In this video we will learn trigonometric ratios of right angled triangle like sin cos tan... etc.For any right triangle, there are six trig ratios: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Here are the formulas for these six trig ratios: Given a triangle, you should be able to identify all 6 ratios for all the angles (except the right angle). Let's start by finding all 6 ratios for angle A ... as the ratio of the sides of a triangle. Also, we were only able to find the value of trig functions of angles upto 90 degrees. But in unit circle definition, the trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x^2 + y^2=1. Learn the basics of trigonometry, such as sine, cosine and tangent, and how to use them to find angles and distances in right-angled triangles. Explore the unit circle, degrees …Learn what are trigonometric ratios, the six basic and important trig ratios in trigonometry, and how to calculate them using formulas and identities. Find out the values of trigonometric ratios for different angles and sides of a right triangle in a table and a formula. Explore the applications of trigonometric ratios in physics, geometry, and calculus. 0. ∞. The Trigonometric Ratios of standard angles are listed below 0°, 30°, 45°, 60° and 90°.The values of Trigonometric Ratios of standard angles are very helpful and important to solve the trigonometric problems. Therefore, it is necessary for you to remember the value of the Trigonometric Ratios of standard angles.Meetings where decisions are made but no one wants to be accountable for them, are the worst meetings. Keep that “action to decision” ratio low by ending every meeting with an “acc...The unit circle definition allows us to extend the domain of sine and cosine to all real numbers. The process for determining the sine/cosine of any angle θ is as follows: Starting from ( 1, 0) . , move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive ...Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. Here the basic trigonometric function of Sin θ = x, can be changed to Sin-1 x = θ. Here x can have values in whole numbers, decimals, fractions, or exponents. For θ = 30° we have θ = Sin-1 (1/2). All the trigonometric formulas can be transformed into ... Trigonometric ratios in right triangles. Google Classroom. β 4 5 3 C B A. Find tan ( β) in the triangle.The reciprocal identities are simply definitions of the reciprocals of the three standard trigonometric ratios: (1.8.1) sec θ = 1 cos θ csc θ = 1 sin θ cot θ = 1 tan θ. Also, recall the definitions of the three standard trigonometric ratios (sine, cosine and tangent): (1.8.2) sin θ = o p p h y p cos θ = a d j h y p tan θ = o p p a d y.Trigonometric Ratios: Cosine. Right triangles have ratios that are used to represent their base angles. Cosine ratios, along with sine and tangent ratios, are ratios of two different sides of a right triangle.Cosine ratios are specifically the ratio of the side adjacent to the represented base angle over the hypotenuse.Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle', and μέτρον (métron) 'measure') [1] is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. Compa ratio is a formula used to assess the competitiveness of an employee’s pay. Learn how to calculate compa ratio. Human Resources | What is WRITTEN BY: Charlette Beasley Publis...Jul 24, 2017 · This math video tutorial provides a basic introduction into trigonometry. It covers trigonometric ratios such as sine, cosine, and tangent. It explains how... Learn how to calculate and use the trigonometric ratios of sine, cosine and tangent based on a right triangle. See examples, graphs, animations and exercises to practice …However, the trigonometric ratios will remain same. Hence we can say that – The trigonometric ratios are same for the same angle. Relation between Trigonometric Ratios. The trigonometric ratios sin θ, cos θ and tan θ of an angle θ are very closely connected by a relation. If any one of them is known, the other two can be easily calculated.21 Dec 2018 ... This tutorial discusses the three primary trigonometric ratios (sin, cos & tan). The video shows how to label sides as hypotenuse, ...Accurate trigonometric ratios for 0°, 30°, 45°, 60° and 90° The trigonometric ratios for the angles 30°, 45° and 60° can be calculated using two special triangles. The trigonometric functions (also called the circular functions) comprising trigonometry are the cosecant cscx, cosine cosx, cotangent cotx, secant secx, sine sinx, and tangent tanx. ... The ratio is defined as . As a result of this definition, the trigonometric functions are periodic with period , so (1)6.5K 443K views 5 years ago Geometry Video Playlist This basic trigonometry video tutorial provides an introduction into trigonometric ratios as it …Learn what trigonometric ratios are and how to calculate them using the sides of a right triangle. Find the values of trigonometric ratios for standard angles and identities for different types of angles.Trigonometric ratios are the ratios between the sides and angles of a right triangle.These ratios are given by the following trigonometric functions of the known angle θ, where a, b and c refer to the lengths of the right triangle’s sides.. Notice that we can label the sides, or legs, of the right triangle as opposite or adjacent, depending on the given …Determine the six trigonometric ratios for angle A in the right triangle. In this tutorial you are shown how to name the sides of any right-angled triangle which is the first stage in solving a question in trigonometry. In this tutorial you are shown how the sin, cos and tan ratios compare two sides of a right-angled triangle.See full list on byjus.com Discover how to measure angles, distances, and heights using trigonometric ratios and the unit circle. Learn how to use sine, cosine, and tangent to solve real-world problems involving triangles and circular motion.What are trigonometric ratios of complementary angles? Ans: Two angles are said to be complementary if their sum equals \({90^ \circ }\). The relationship between the acute angle and the lengths of sides of a right-angled triangle is expressed by the trigonometric ratios. So, the trigonometric ratio of complementary angle is a …1 Introduction. You have probably met the trigonometric ratios cosine, sine, and tangent in a right angled triangle, and have used them to calculate the sides and angles of those triangles. In this booklet we review the definition of these trigonometric ratios and extend the concept of cosine, sine and tangent.This is also the relationship between all the other cofunctions in trigonometry: tan (θ)=cot (90°-θ), sec=csc (90°-θ). One other way to think about the relationship between a function and its cofunction is to think about the unit circle: your x-distance is described by cos (θ), and your y-distance described by sin (θ). Trigonometry examines the relationship between the sides of a triangle, more specifically, right triangles. A right triangle has a 90° angle. The equations and ratios that describe the relationship between the sides of a triangle and its angles are trigonometric functions. In this particular article, we're going to explain one specific ratio: "cos" or cosine.CosY = 0.30. This is where the Inverse Functions come in. If we know that CosY = 0.30, we're trying to find the angle Y that has a Cosine 0.30. To do so: -Enter 0.30 on your calculator. -Find the Inverse button, then the Cosine button (This could also be the Second Function button, or the Arccosine button).
Learn the six trigonometric ratios of a right angle triangle and how to calculate them using formulas and mnemonics. See solved examples and a table of values for different …. Nike calm mule
Exercise 5.2e. ★ Given right triangle where the right angle is angle C in each figure below, (a) Label the remaining sides and angles. (b) Designate the hypotenuse, adjacent side or opposite side to angle A. Determine the trigonometric ratios for (c) sinA, (d) cosA, (e) tanA, (f) secA, (g) cscA, (h) cotA. Give simplified exact answers ...Trigonometry values of different ratios, such as sine, cosine, tangent, secant, cotangent, and cosecant, deal with the measurement of lengths and angles of the right-angle triangle. The values of trigonometric functions for 0°, 30°, 45°, 60° and 90° are commonly used to solve trigonometry problems.. Trigonometry values are all about the study of standard …The values of trigonometrical ratios of standard angles are very important to solve the trigonometrical problems. Therefore, it is necessary to remember the ...6.5K 443K views 5 years ago Geometry Video Playlist This basic trigonometry video tutorial provides an introduction into trigonometric ratios as it …A debt-to-income ratio (DTI) is a tool lenders use to determine whether they believe you can manage additional debt. Read to find what's considered a good DTI. Get top content in o...Find trigonometric ratios using the Pythagorean theorem. tan ( θ) = 7 24 for an acute angle θ . Find sec ( θ) without the use of a calculator. Stuck? Use a hint. Loading... Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the ... Review all six trigonometric ratios: sine, cosine, tangent, cotangent, secant, & cosecant. What are the trigonometric ratios? C B A opposite adjacent hypotenuse Want to learn …Step 2: Label the sides of the triangle according to the ratios of that special triangle. 30 ∘ 60 ∘ x 3 x 2 x. Step 3: Use the definition of the trigonometric ratios to find the value of the indicated expression. sin ( 30 ∘) = opposite hypotenuse = x 2 x = 1 x 2 x = 1 2. Note that you can think of x as 1 x so that it is clear that x 2 x ...31 Oct 2023 ... Trigonometric ratios formula for class 10 math students. Check out the complete series on TRIGONOMETRY using Animation and visual tools: ...Three Functions, but same idea. Right Triangle. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.. Before getting stuck into the functions, it helps to give a name to each side of a right triangle: The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are …About this unit. In this unit, you'll explore the power and beauty of trigonometric equations and identities, which allow you to express and relate different aspects of triangles, circles, and waves. You'll learn how to use trigonometric functions, their inverses, and various identities to solve and check equations and inequalities, and to ... Find trigonometric ratios using the Pythagorean theorem. tan ( θ) = 7 24 for an acute angle θ . Find sec ( θ) without the use of a calculator. Stuck? Use a hint. Loading... Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the ...The right equipment, training, and running route isn't always enough to ensure a good run, especially if your breathing is all over the map. To help stabilize yours, try incorporat...This math video tutorial provides a basic introduction into trigonometry. It covers trigonometric ratios such as sine, cosine, and tangent. It explains how...All the trigonometric identities are based on the six trigonometric ratios. They are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from ....