Exercise Set 6.2: Double-Angle and Half-Angle Formulas
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1 Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin ( x sin ( x = π (a Evaluate cos π (b Evaluate π π (c Is cos = (d Graph f ( x cos( x g ( x cos ( x = and = on the same set of axes (e Is cos ( x cos ( x = π 3 (a Evaluate tan π (b Evaluate π π (c Is tan = (d Graph f ( x tan ( x g ( x tan ( x = and = on the same set of axes (e Is tan ( x tan ( x = 4 Derive the formula for sin ( θ by using a sum formula on sin ( θ + θ Derive the formula for cos ( θ by using a sum formula on cos ( θ + θ Derive the formula for tan ( θ by using a sum formula on tan ( θ + θ 7 The sum formula for cosine yields the equation cos θ = cos θ sin θ To write cos( θ strictly in terms of the sine function, (a Using the Pythagorean identity ( θ + ( θ =, solve for cos sin 1 cos θ (b Substitute the result from part (a into the above equation for cos( θ 8 The sum formula for cosine yields the equation cos( θ = cos ( θ sin ( θ To write cos( θ strictly in terms of the cosine function, (a Using the Pythagorean identity ( θ + ( θ =, solve for cos sin 1 sin θ (b Substitute the result from part (a into the above equation for cos( θ Answer the following 9 Suppose that cos (a sin α (b cos α (c tan α 10 Suppose that tan (a sin α (b cos α (c tan α 11 Suppose that sin (a sin α (b cos α (c tan α 1 α = and 3 π < α < π π α = and π < α < 4 π α = and < α < π
2 Exercise Set : Double-Angle and Half-Angle Formulas 1 Suppose that tan 3 (a sin α (b cos α (c tan α α = and 3 π < α < π 4 sin 7 cos ( 7 tan 41 tan 41 cos ( Simplify each of the following expressions as much as possible without a calculator 13 sin ( 1 cos( β cos 1 ( α ( α tan 3 tan cos β β sin 1 cos 34 1 π 1 sin 1 tan ( tan sin ( x 7π 7π 19 0 sin ( 3 cos( 3 The formulas for sin x and x cos both contain a ± sign, meaning that a choice must be made as to whether or not the sign is positive or negative For each of the following examples, first state the quadrant in which the angle lies Then state whether the given expression is positive or negative (Do not evaluate the expression 9 (a cos( 10 (b sin ( 7 30 (a sin ( 1 (b cos( cos sin 1 tan 1 π sin (a (b 3 (a (b 1π 1 7π 19π 1
3 Exercise Set : Double-Angle and Half-Angle Formulas In the text, tan is defined as: ( s sin s tan = 1 + cos The following exercises can be used to derive this formula along with two additional formulas for tan 33 (a Write the formula for (b Write the formula for (c Derive a new formula for ` using the sin ( θ s identity tan ( θ =, where θ = cos θ Leave both the numerator and denominator in radical form Show all work 34 (a This exercise will outline the derivation for: sin ( s = In exercise 33, it was 1 + cos ( s discovered that ( s cos s = ± 1 + cos Rationalize the denominator by multiplying both the numerator and denominator by 1+ cos( s Simplify the expression and write the result for (b A detailed analysis of the signs of the trigonometric functions of s and s in various quadrants reveals that the ± symbol in part (a is unnecessary (This analysis is lengthy and will not be shown here Given this fact, rewrite the formula from part (a without the ± symbol (c How does this result from part (b compare with the formula given in the text for 3 (a In exercise 33, it was discovered that ( s cos s = ± 1 + cos Rationalize the numerator by multiplying both the numerator and denominator by cos ( s Simplify the expression and write the new result for (b A detailed analysis of the signs of the trigonometric functions of s and s in various quadrants reveals that the ± symbol in part (a is unnecessary (This analysis is lengthy and will not be shown here Given this fact, rewrite the formula from part (a without the ± symbol This gives yet another formula which can be used for (c Use the results from Exercises 33-3 to write the three formulas for Which formula seems easiest to use and why Which formula seems hardest to use and why 3 (a In the text, is defined as: ( s sin s = 1 + cos Multiply both the numerator and denominator of the right-hand side of the Then simplify to obtain a formula for Show all work equation by 1 cos( s (b How does the result from part (a compare to the identity obtained in part (b of Exercise 39
4 Exercise Set : Double-Angle and Half-Angle Formulas Answer the following SHOW ALL WORK Do not leave any radicals in the denominator, ie rationalize the denominator whenever appropriate 37 (a Find cos( 7 by using a sum or difference (b Find cos( 7 by using a half-angle (c Enter the results from parts (a and (b into a calculator and round each one to the nearest hundredth Are they the same 38 (a Find sin ( 1 by using a sum or difference (b Find sin ( 1 by using a half-angle (c Enter the results from parts (a and (b into a calculator and round each one to the nearest hundredth Are they the same 39 (a Find sin ( 11 by using a half-angle (b Find cos( 11 by using a half-angle (c Find tan ( 11 by computing sin ( 11 cos ( 11 (d Find tan ( 11 by using a half-angle 40 (a Find sin ( 10 by using a half-angle (b Find cos( 10 by using a half-angle (c Find tan ( 10 by computing sin 10 cos 10 (d Find tan ( 10 by using a half-angle Find the exact value of each of the following by using a half-angle Do not leave any radicals in the denominator, ie rationalize the denominator whenever appropriate 41 (a (b (c 4 (a (b (c 43 (a (b (c (a sin ( 17 (b cos( 17 (c tan ( 17 4 (a sin ( 8 (b cos( 8 (c tan ( 8 4 (a (b (c 1 1 1
5 Exercise Set : Double-Angle and Half-Angle Formulas Answer the following 47 If cos θ = and 3 π < θ < π, 9 (a Determine the quadrant of the terminal side of θ (b Complete the following: θ < < (c Determine the quadrant of the terminal side of θ (d Based on the answer to part (c, determine the sign of (e Based on the answer to part (c, determine the sign of (f Find the exact value of (g Find the exact value of (h Find the exact value of 48 If sin 1 3π θ = and π < θ <, (a Determine the quadrant of the terminal side of θ (b Complete the following: θ < < (c Determine the quadrant of the terminal side of θ (d Based on the answer to part (c, determine the sign of (e Based on the answer to part (c, determine the sign of (f Find the exact value of (g Find the exact value of (h Find the exact value of 49 If tan 7 π θ = and < θ < π, 3 (a Find the exact value of (b Find the exact value of (c Find the exact value of 0 If cos θ = and 3 π < θ < π, (a Find the exact value of (b Find the exact value of (c Find the exact value of Prove the following 1 ( x ( x cos sin ( x ( x = tan ( x ( x cos 3 sin 3 cos sin ( x = 3 1 sin ( x 1 + sin ( x = cos( x 4 cos 4 ( x sin 4 ( x = cos ( x x csc cot = ( x ( x x 1+ tan = x tan sec( θ
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