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Chapter 7 Section 7.1 Fundamental Identities Trigonometric Relations The six trigonometric functions are related in many different ways. Several of these are quite useful for solving different problems, finding values for the trigonometric functions or solving trigonometric equations. Fundamental Trigonometric Identities: (They form the building blocks for many other identities.) Reciprocal Identities: 1 csc x sin x 1 sec x cos x 1 cot x tan x sin x tan x cos x cos x cot x sin x Pythagorean Identities: sin x cos x 1 tan x 1 sec x 1 cot x csc x 2 2 2 2 2 2 Even/Odd Identities: sin( x) sin x cos( x) cos x tan( x) tan x Co-function Identities: sin 2 x cos x tan 2 x cot x sec2 x csc x cos2 x sin x cot 2 x tan x csc2 x sec x Simplifying Trigonometric Expressions It is often useful to be able to simplify a trigonometric expression. For example, when you are trying to solve an equation that involves trigonometric functions. There are many ways to do this but the two that are used very often are: 1. Change to sines and cosines. 2. Combine fractions and expressions where possible. Here are some examples. (Notice these are expressions! (i.e. there is no equal sign).) Simplify Simplify tan x csc x sin x 1 cot x sin x 1 cos x sin x 1 cos x sec x 2 2 2 2 sin x csc x 1 2 sin x 2 sin x 1 Simplify csc x sin x cot x 1 sin x sin x cos x sin x sin1 x sin x sin x cos x sin x sin x 1 sin x cos x 2 2 cos x cos x cos x Find the value for sin 𝜃 if the sec 𝜃 = and tan 𝜃 < 0. 7 2 First need to figure out the sign. Since secant is positive 𝜃 is in either quadrants I or IV, and tangent is negative 𝜃 is either in II or IV. The angle 𝜃 must be in quadrant IV where sine is negative. 1 2 cos 𝜃 = = sec 𝜃 7 sin2 𝜃 + cos2 𝜃 = 1 4 45 sin 𝜃 = 1 − cos 𝜃 = 1 − = 49 49 2 2 45 45 sin 𝜃 = − =− 49 7 Find the values of all trigonometric functions if −8 cot 𝜃 = and sin 𝜃 > 0. 5 Sine positive in I or II and Cotangent negative in II or IV so 𝜃 is in quadrant II. Reciprocal −5 tan 𝜃 = 8 89 sec 𝜃 = − sin 𝜃 = 8 89 5 Pythagorean csc 2 𝜃 = 1 + cot 2 𝜃 64 89 2 csc = 1 + = 25 25 89 89 csc 𝜃 = ± = 5 5 cos 2 𝜃 = 1 − sin2 𝜃 25 64 2 cos = 1 − = 89 89 8 8 cos 𝜃 = ± =− 89 89 Any one trigonometric function can be expressed as some sort of combination of another trigonometric function. The Identities allow you to do this conversion. Example Write the csc 𝑥 in terms of the cos 𝑥. (This means find out what the csc 𝑥 is equal to but only using cos 𝑥) 1 csc 𝑥 = sin 𝑥 Using the Pythagorean Identity sin 𝑥 can be written in terms of cos 𝑥. sin2 𝑥 = 1 − cos2 𝑥 sin 𝑥 = ± 1 − cos2 𝑥 Substitute back: csc 𝑥 = 1 ± 1 − cos2 𝑥