Chapter 4 Using Fundamental Identities Section 4.1 4.1 USING FUNDAMENTAL IDENTITIES Fundamental Trigonometric Identities Reciprocal Identities csc x sec x cot x Quotient Identities tan x cot x Pythagorean Identities (?,? ) x Cofunction Identities x sin x tan x sec x cos x cot x csc x Even/Odd Identities sin( x) cos( x) tan( x) 1
Chapter 4 Using Fundamental Identities Section 4.1 Using the Fundamental Identities Example #1: Use the given values to evaluate the remaining trigonometric functions using identities. 4 3 3 3 (a) sin x,cot x (b) tan x, cos x 5 4 3 Example #: Use the fundamental identities to find the exact values of the remaining trig functions of x. 1 (a) tan x, cos x 0 (b) sec 3, tan 0 Example #3 & 4: Use the fundamental identities to simplify each expression. cos x sin x cot x csc x 1 3a. 3b. 4a. cos x csc x 1 4b. csc x 1 sin sin x x
Chapter 4 Using Fundamental Identities Section 4.1 Example #5: Use fundamental identities and appropriate algebraic operations to simplify 1 cos 1 three different ways. cos Example #6: Use fundamental identities and appropriate algebraic operations to simplify 1 cos 1 two different ways. tan Example #7: Use fundamental identities and appropriate algebraic operations to simplify 1 two different ways. cot Example #8: Simplify each expression. (a) sin x (cos x 1) (b) sec x(1 sin x) (c) cot xcos( x) (d) sec x tan x sec x \ 3
Chapter 4 Using Fundamental Identities Section 4. VERIFYING TRIGONOMETRIC IDENTITIES Rules for Verifying Trigonometric Identities Since we are proving left side = right side, you cannot solve proofs like you solve equations. Therefore, you cannot use algebraic steps even if they are done equally to both sides. To verify an identity, start with the expression on one side and convert it into the expression on the other side using known identities and/or algebra. You may make changes to either side INDEPENDENTLY. Never use an identity (or any alternate version of one) to prove itself! Guidelines for Verifying Trigonometric Identities 1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 3. Look for opportunities to use algebraic operations such as multiplying or canceling, factoring the expression, combining fractions or splitting fractions, squaring a binomial, or creating a monomial denominator. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Try something! At each step, keep the other side of the identity in mind. Even making an attempt that leads to a dead end gives insight. Examples #1 & #: Verify each trigonometric identity. 1a) sec( x) secx 1b) cot(- x)cos(- x) sin x csc x a) cot x cos x sin x csc x b) sec x tan xsin x cos x 1
Chapter 4 Using Fundamental Identities Section 4. Examples #3 & #4: Verify each trigonometric identity. tan x -1 3a) 1- cos 1 tan x x 3b) tan x 7 cot x sec x 6 tan x 5 cot x sec x 4 4a) 1 sin x cos x sec x cos x 1 sin x 4b) cos x 1 csc x 3 sin x 1 cos x cos x Example #5: Verify the trigonometric identity sec x tan x from left to right, then from right to left. 1- sin x
Chapter 4 Using Fundamental Identities Section 4. Extra Practice: Verify each trigonometric identity. a) cos sin cos 1 b) 1 1 tan cot tan cot c) sec cot 1 d) (1 sin z)[1 sin( z)] cos z f) cos x cos y sin x sin y 0 sin x sin y cos x cos y g) 6 4 5 3 sec x(sec x tan x) sec x(sec x tan x) sec x tan x h) cot x csc x 1 csc x 1 cot x i) 4 5 sin x(1 cos x cos x) sin x 3
Chapter 4 Using Fundamental Identities Section 4. Extra Practice (continued): Verify each trigonometric identity. j) cos x sec x cos x sin x k) sec x tan x tan x cos x sin x sec x tan 1 sec l) csc 1 sec tan m) 1 csc x sec x cot x cos x n) sin cos cos sin sec csc o) sin cos sin x cos x 1 sin x 1 sin x cos x 1 cos x 4
Chapter 4 Using Fundamental Identities Section 4.3 4.3 SUM & DIFFERENCE IDENTITIES We are going to derive all the sum and difference formulas. cos( x y)? cos( x y) sin( x y) sin( x y) tan( x y) tan( x y) 1
Chapter 4 Using Fundamental Identities Section 4.3 Sum and Difference Formulas sin( x y) sin( x y) cos( x y) cos( x y) tan( x y) tan( x y) Example #1: Simplify each expression using sum and difference formulas. a) sin( x 30 ) b) tan x 4 c) cos 5 x 3 d) cot( x 135 ) Example #: Find the exact value of the trigonometric function. a) cos b) tan 75 1 Example #3: Use the sum and difference formulas to write the expression as the sine, cosine, or tangent of an angle. a) cos 0 cos30 sin 0 sin 30 b) sin cos cos sin 9 10 9 10 c) 7 tan tan 1 6 7 1 tan tan 1 6
Chapter 4 Using Fundamental Identities Section 4.3 Evaluating a Trigonometric Expression 5 3 Example #4a: Find the exact value of cos( u v) given that sin u, where 0 u and cos v, where v. 13 5 4 Example #4b: Find the exact value of sin( u v) given that sin u, where u and 5 8 3 cos v, where v. 17 Cofunction Identity (Again!) Example #5: Use the sum and difference formulas to prove the following cofunction identities. a) sin x b) cos x c) tan x d) cot x 3
Chapter 4 Using Fundamental Identities Section 4.3 Verifying Trigonometric Identity (Didn t we do these already?...) Example #6: Verify the identity. a) sin( x ) sin x b) tan x tan x 1 tan x c) 1 tan tan 4 1 tan d) sin( x y)sin( x y) sin x sin y 4
Chapter 4 Using Fundamental Identities Section 4.4 4.4 DOUBLE-ANGLE & HALF-ANGLE IDENTITIES Double-Angle Formulas sin x cos x tan x Example #1a: Suppose 15 cos u and 0 u, find the exact values of sin u, cos u, and tan u using the double- 17 angle formulas. Example #1b: Suppose 3 cot u 6 and u, find the exact values of sin u, cos u, and tan u using the double- angle formulas. 1
Chapter 4 Using Fundamental Identities Section 4.4 Half-Angle Formulas Derive half-angle identities by substituting u = x/ into the double-angle identities, then solving for the trig function with the x/ angle. cos u 1 sin u cos u cos u 1 To obtain a half-angle identity for the tangent function, we use the quotient identity and the half-angle formulas for sine and cosine: x sin x tan x cos Half-Angle Formulas x 1 cos x sin x 1 cos x cos tan x 1 cos x sin x sin x 1 cos x The signs of sin x x x and cos depend on the quadrant in which lies. Example #: Find the exact value of the following. a) cos105 b) tan165 c) 11 sin 1 d) 3 cot 8
Chapter 4 Using Fundamental Identities Section 4.4 Example #3a: Find the exact values of sin u, cos u, and tan u 7 if cos u,0 u. 5 Example #3b: Find the exact values of sin u, cos u, and tan u 3 3 if sin u, u. 4 Example #4: True or false. Explain why. cos cos Example #5: Verify the identity algebraically. a) sec x sec x b) sec x x tan x sin x sin tan x 3