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10/28/2014 Introduction We will learn how to use the fundamental Using Fundamental Identities Precalculus 5.1 identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions. 3. Develop additional trigonometric identities. 4. Solve trigonometric equations. Introduction Introduction Example 1 Example 2 Use the values sin x 12 and cos x 0 to fine the values of all six trigonometric functions. cont’d Simplify cos 2 x csc x csc x 1 10/28/2014 Example 3 Factor each expression. a) 1 cos 2 x b) 2 csc 2 x 7 csc x 6 Example 5 Simplify csc t cos t cot t Example 4 Factor sec 2 x 3 tan x 1 Example 6 Perform the addition and simplify. 1 1 1 sin 1 sin Example 7 Rewrite the statement so that it is NOT in fractional form. cos 2 y 1 sin y Example 8 Use the substitution x 5 sin ,0 2 to write 25 x 2 as a trigonometric function of . 2 10/28/2014 Example 9 Rewrite ln sec ln cot as a single logarithm and simplify the result. Verifying Trigonometric Identities Precalculus 5.2 Introduction In this section, you will study techniques for verifying trigonometric identities. Remember that a conditional equation is an equation that is true for only some of the values in its domain. For example, the conditional equation sinx = 0 Introduction On the other hand, an equation that is true for all real values in the domain of the variable is an identity. For example, the familiar equation Identity sin2x = 1 – cos2x Conditional equation is true for all real numbers x. So, it is an identity. is true only for x = n, where n is an integer. When you find these values, you are solving the equation. Verifying Trigonometric Identities Although there are similarities, verifying that a trigonometric equation is an identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities, and the process is best learned by practice. Example 1 Verify the identity sin 2 cos 2 1 cos 2 sec 2 3 10/28/2014 Example 2 Verify the identity 1 1 2 csc 2 1 cos 1 cos Example 4 Verify the identity csc x sin x cos x cot x Example 6 Verify the identity 2 tan 1 cos 1 sec cos Example 3 Verify the identity sec 2 x 1 sin 2 x 1 sin 2 x Example 5 Verify the identity csc cot sin 1 cos Example 7 Verify each identity a) tan 3 x tan x sec 2 x tan x b) cos 3 x sin 4 x (sin 4 x sin 6 x ) cos x c) cot 4 x csc x cot 2 x (csc 3 x csc x ) 4 10/28/2014 Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Solving Trigonometric Equations Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function in the equation. Precalculus 5.3 For example, to solve the equation 2 sin x = 1, divide each side by 2 to obtain Introduction Introduction To solve for x, note in Figure 5.6 that the equation has solutions x = /6 and x = 5 /6 in the interval [0, 2). Moreover, because sin x has a period of 2, there are infinitely many other solutions, which can be written as and General solution where n is an integer, as shown in Figure 5.6. Figure 5.6 Introduction Another way to show that the equation many solutions is indicated in Figure 5.7. Example 1 has infinitely Solve sin x 2 sin x Any angles that are coterminal with /6 or 5 /6 will also be solutions of the equation. When solving trigonometric equations, you should write your answer(s) using exact values rather than decimal approximations. Figure 5.7 5 10/28/2014 Example 2 Solve 4 sin 2 x 3 0 Example 4 Find all solutions of the equation on the interval 0,2 . Example 3 Solve sin 2 x 2 sin x Example 5 Solve 3 sec 2 x 2 tan 2 x 4 0 2 sin 2 x 3 sin x 1 0 Example 6 Find all of the solutions of the equation on the interval 0,2 . Example 7 Find all the solutions of sin 2 x 3 2 0 sin x 1 cos x 6 10/28/2014 Example 8 Find all the solutions of tan 2x 1 0 Example 9 Find all the solutions of 4 tan 2 x 5 tan x 6 Using Sum and Difference Formulas Sum & Difference Formulas Precalculus 5.4 Example 1 Find the exact value of cos 12 Example 2 Find the exact value of sin 75 7 10/28/2014 Example 3 Find the exact value of cos(u v ) given sin u where 0 u 2 , and cos v 45 , where 2 Example 4 7 25 , Write sin(arctan 1 arccos x ) as an algebraic expression. v . Example 5 Prove the cofunction identity sin x 2 cos x Example 6 Simplify each expression a) sin 32 b) tan 4 Example 7 Find all of the solutions of the equation on the interval 0,2 . sin x 2 sin x 32 1 Example 8 Verify that cos x h cos x cosh 1 sinh cos x sin x h h h 8 10/28/2014 Multiple-Angle Formulas Multiple-Angle & Product-to-Sum Formulas You should learn the double-angle formulas because they are used often in trigonometry and calculus. Precalculus 5.5 Example 1 Solve cos 2 x cos x 0 Example 3 Use sin u 35 ,0 u 2 , to find sin 2u , cos 2u , and tan 2u. Example 2 Use a double angle formula to rewrite the equation g ( x ) 3 6 sin 2 x. Then sketch the graph of the equation over the interval 0,2 . Example 4 Derive a triple-angle formula for cos 3 x. 9 10/28/2014 Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. Example 5 shows a typical power reduction that is used in calculus. Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by replacing u with u / 2. The results are called half-angle formulas. Example 7 Find all solutions of cos 2 x sin 2 interval 0,2 . Example 5 Rewrite tan 4 x as a quotient of first powers of the cosines of multiple angles. Example 6 Find the exact value of cos 105 Product-to-Sum Formulas x 2 on the Each of the following product-to-sum formulas can be verified using the sum and difference formulas. Product-to-sum formulas are used in calculus to evaluate integrals involving the products of sines and cosines of two different angles. 10 10/28/2014 Example 8 Rewrite sin 5 cos 3 as a sum or difference. Example 9 Find the exact value of sin 195 sin 105 Example 11 Verify the identity sin 6 x sin 4 x tan 5 x cos 6 x cos 4 x Product-to-Sum Formulas Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the following sum-to-product formulas. Example 10 Solve sin 4 x sin 2 x 0 Example 12 Ignoring air resistance, the range of a projectile fired at an angle with the horizontal and with an initial velocity of v0 feet per second is given by r 161 v0 2 sin cos , where r is the horizontal distance (in feet) that the projectile will travel. A place kicker for a football team can kick a football from ground level with an initial velocity of 78 feet per second. 11 10/28/2014 Example 12 cont’d a) At what angle must the player kick the football so that it travels 188 feet? b) For what angle is the horizontal distance the football travels a maximum? 12