CHAPTER 5: Analytic Trigonometry

Transcription

1 ) (Answers for Chapter 5: Analytic Trigonometry) A.5. CHAPTER 5: Analytic Trigonometry SECTION 5.: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) csc( x) sin x Reciprocal ID tan( x) tan( x) tan π x cot x sin x cos x Reciprocal ID Quotient ID cot ( x ) Cofunction ID cos x sin π x Cofunction ID sin x cos x tan x sin x cos x tan x sin ( x) + cos x tan ( x) + sec x + cot x Even / Odd (Negative-Angle) ID Even / Odd (Negative-Angle) ID Even / Odd (Negative-Angle) ID Pythagorean ID Pythagorean ID Pythagorean ID csc x ) a) sec( x); b) sec ( θ ); c) ; d) csc 4 ( x); e) sin( t); f) sin( α ) 3) a) 4 cos( θ ); b) 6sec( θ ); c) 3tan( θ ) SECTION 5.: VERIFYING TRIGONOMETRIC IDENTITIES ) Solutions will vary.

2 (Answers for Chapter 5: Analytic Trigonometry) A.5. SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS ) a) x x = π 3 + π n or x = π π. In [ 0, π ): 3 3, π 3 b) θ θ = ± 3π, or, equivalently, 4 θ θ = 3π 4 + π n or θ = 5π 4 + π n ( n ). In [ 0, π ): c) No real solutions; the solution set is. No real solutions in 0, π [ ).. 3π 4, 5π 4 d) u u = 3π 3π. Solutions in [ 0, π ) :. e) u u = π + π n ( n π ). Solutions in [ 0, π ) :, 3π. f) u u = 7π π + π n or u =, or, equivalently, 6 6 u u = 7π 6 + π n or u = π 6 + π n ( n 7π ). In [ 0, π ) : 6, π. 6 g) x x = ± π, or, equivalently, 3 x x = π 3 + π n or x = 5π 3 + π n ( n π ). In [ 0, π ) : 3, 5π. 3 h) No real solutions; the solution set is. No real solutions in 0, π i) x x = π 6 + π n ( n ). Solutions in 0, π j) θ θ = π + π n ( n ). Solutions in 0, π [ ) : [ ) : π 6, 7π. 6 π, 3π [ ). k) θ θ = ± π 6 + π n ( n ), or, equivalently, θ θ = π 6 + πn or θ = 5π 6 + πn π. In [ 0,π ): 6, 5π 6, 7π 6,π 6...

3 (Answers for Chapter 5: Analytic Trigonometry) A.5.3 l) θ θ = π n or θ = 3π 4 + π n ( n ), or, equivalently, θ θ = π n or θ = π 4 + π n ( n ). In [ 0, π ) : 0, 3π 4, π, 7π 4. m) x x = π 6 + π n or x = π + π n or x = 5π 6. π Solutions in [ 0, π ) : 6, π, 5π 6. n) x x = π π n + π n or x =, by rotational symmetry. Less 3 efficiently: x x = π + π n or x = π n or x = ± π 3. Solutions in [ 0, π ) : 0, π, π 3, 4π 3, 3π. o) x x = ± π + π n. The following form may be more useful for later: x x = π + π n or x = 5π + π n. π Solutions in [ 0, π ) :, 5π, 7π, π,3π, 7π, 9π, 3π. p) x x = π 6 + π n π. In [ 0, π ) : 3 6, 5π 6, 3π. q) x x = ± π 9 + π n. The following form may be more useful for 3 later: x x = π 9 + π n or x = π π n. Solutions in [ 0, π ) : 3 π 9, π 9, 4π 9, 5π 9, 7π 9, 8π 9, 0π 9, π 9, 3π 9, 4π 9, 6π 9, 7π 9. ) a) { arctan, π + arctan }; equivalently, { tan, π + tan }. b) Approximately: {.07, 4.49}. (Make sure your calculator is in radian mode.) c) { x x = arctan + π n }, or { x x = tan + π n }.

4 (Answers for Chapter 5: Analytic Trigonometry) A.5.4 3) ) a) Solutions in [ 0, π ) : arccos 5, π + arccos 5. Equivalent forms: cos 5, π + cos 5, π arccos 5, π + arccos 5, and arccos 5, π arccos 5. b) Approximately: {.77, 4.5}. (Make sure your calculator is in radian mode.) c) x x = ± arccos 5 + π n ( n ), or, equivalently, x x = ± cos 5 + π n ( n ), or, equivalently, x x = ± arccos 5 + ( n + )π. SECTIONS 5.4 and 5.5: MORE TRIGONOMETRIC IDENTITIES Left Side Right Side Type of Identity (ID) sin u sin u + v cos( v) + cos( u) sin( v) Sum ID cos( u + v) cos( u) cos( v) sin( u) sin( v) Sum ID tan( u + v) tan u + tan( v) tan( v) tan u Sum ID sin( u v) sin( u) cos( v) cos( u) sin( v) Difference ID cos( u v) cos( u) cos( v) + sin( u) sin( v) Difference ID tan( u v) tan u tan( v) tan( v) + tan u Difference ID sin( u) sin( u) cos( u) Double-Angle ID

5 ) 3) 4) a) (Answers for Chapter 5: Analytic Trigonometry) A.5.5 Left Side Right Side Type of Identity (ID) cos cos( u) ( u) sin ( u), sin ( u), and Double-Angle ID cos u (write all three versions) tan( u) sin ( u) cos ( u) sin θ cos θ tan θ cos u + cos u ( u) tan u tan or or cos( u) + cos( u) cos θ ± (Choose the sign appropriately.) + cos θ ± (Choose the sign appropriately.) ± ; b) cos θ + cos θ cos θ = = sin θ sin θ + cos θ (Choose the sign appropriately.) ) a) ; b) ; c) ; d) 3 6) cos θ 7) 8) 6 tan 4x Double-Angle ID Power-Reducing ID (PRI) Power-Reducing ID (PRI) Half-Angle ID Half-Angle ID Half-Angle ID (write all three versions) ; c) 3 + (rationalize the denominator in ). a) Hint: Use a Sum Identity. b) Hints: Use a Double-Angle Identity and a Pythagorean Identity. c) Hints: Use the Sum Identities for sine and cosine, and then divide the numerator and the denominator by cos( u)cos( v).

6 (Answers for Chapter 5: Analytic Trigonometry) A ) a) All real solutions: x x = π 5π + π n or x = + π n ( n ). π Solutions in [ 0, π ) :, 5π, 3π, 7π b) All real solutions: x x = ± π + π n or x = π 3, or, equivalently, x x = π 3 + π n or x = 4π + π n or x = π 3. π Solutions in [ 0, π ) : 3, π, 4π 3 c) All real solutions: x x = π n or x = ± π 3, or, equivalently, x x = π n or x = π 3 + π n or x = 5π 3, or, equivalently, Solutions in 0, π 0) x x ) cos 4 ( x) = ) a) cos θ 3 8 x x = π n or x = π 3 + π n 3 [ ) : 0, π 3, π, 5π cos ( 8θ ) cos ( x ) + 8 b) cos( 4α )cos( α ); c) sin( x)cos( x) d) sin 9θ e) cos 3x h) sin 9α sin( θ ) cos( 5x) sin( α ) cos ( 4x ). + cos ( 8θ ), which is simplified from cos θ ; + sin ( θ ), which is simplified from sin 9θ ; ; f) sin( 4x)sin( 3x); g) cos( 5α )sin( 3α ) ;

7 (Answers for Chapter 6: Additional Topics in Trigonometry) A.6. CHAPTER 6: Additional Topics in Trigonometry SECTION 6.: THE LAW OF SINES ) a) 35.0 m; b).0 m; c) 37 m ) a) ft; b) 4.86 ft; c) 0,37 ft SECTION 6.: THE LAW OF COSINES ) a) 5.8 ; b) 40. ; c) No (that would violate the Triangle Inequality); d) 496 ft ) 3.8 mi SECTION 6.3: VECTORS IN THE PLANE ) a), 3 or m, 3 m ; b) 3 m; c) 56.3 ) a) 5, 3 or 5 m, 3 m ; b) 34 m; c) ) a) v 3 v 3v b) 4, 3 c) d) e) 5, 5

8 (Answers for Chapter 6: Additional Topics in Trigonometry) A.6.. 4) 8.0 ft, 8.9 ft 5) a) 9 9, ; b).8 ; c) 8 9 9, ) a) ; b) 70 7, ) Yes 8) No (they point in opposite directions) 9) a) 0.3 mph; b) 8.3 mph ) 4 SECTION 6.4: VECTORS AND DOT PRODUCTS ) a) scalar; b) vector; c) undefined; d) scalar; e) undefined; f) undefined 3) 0 4) Hint: v + w = ( v + w) ( v + w). 5) v + w 6) The Pythagorean Theorem 7) 9.7 ; acute 8) 67.7 ; obtuse 9) Hint: Find the angle between the vectors BA and BC. 0) a) 0 ; b) 80 ; c) 90 ) Yes ) No 3) 0 and 4) Hint: Use the formula: cos( θ ) = v w v w. 5) 4 7 7