If [tex]$\sin \theta=\frac{4}{9}, \theta$[/tex] in quadrant II, find the exact value of
(a) [tex]$\cos \theta$[/tex]
(b) [tex]$\sin \left(\theta+\frac{\pi}{6}\right)$[/tex]
(c) [tex]$\cos \left(\theta-\frac{\pi}{3}\right)$[/tex]
(d) [tex]$\operatorname { t a n } \left(\theta+\frac{\pi}{4}\right)$[/tex]
Which of the following is the correct answer?
A. B. C.
Answer (2)
Determine cos θ using the Pythagorean identity and the quadrant information: cos θ = − 9 65 .
Apply the angle sum formula for sine to find sin ( θ + 6 π ) = 18 4 3 − 65 .
Apply the angle difference formula for cosine to find cos ( θ − 3 π ) = 18 4 3 − 65 .
Apply the angle sum formula for tangent to find tan ( θ + 4 π ) = 49 81 − 8 65 .
cos θ = − 9 65 , sin ( θ + 6 π ) = 18 4 3 − 65 , cos ( θ − 3 π ) = 18 4 3 − 65 , tan ( θ + 4 π ) = 49 81 − 8 65
Explanation
Problem Setup We are given that sin θ = 9 4 and θ is in quadrant II. We need to find the exact values of cos θ , sin ( θ + 6 π ) , cos ( θ − 3 π ) , and tan ( θ + 4 π ) .
Finding cos(theta) Since θ is in quadrant II, cos θ < 0 . Using the Pythagorean identity sin 2 θ + cos 2 θ = 1 , we have cos 2 θ = 1 − sin 2 θ = 1 − ( 9 4 ) 2 = 1 − 81 16 = 81 81 − 16 = 81 65 . Therefore, cos θ = − 81 65 = − 9 65 .
Finding sin(theta + pi/6) Using the angle sum formula for sine, we have sin ( θ + 6 π ) = sin θ cos 6 π + cos θ sin 6 π = 9 4 ⋅ 2 3 + ( − 9 65 ) ⋅ 2 1 = 18 4 3 − 18 65 = 18 4 3 − 65 .
Finding cos(theta - pi/3) Using the angle difference formula for cosine, we have cos ( θ − 3 π ) = cos θ cos 3 π + sin θ sin 3 π = ( − 9 65 ) ⋅ 2 1 + 9 4 ⋅ 2 3 = − 18 65 + 18 4 3 = 18 4 3 − 65 .
Finding tan(theta + pi/4) We have tan θ = c o s θ s i n θ = − 9 65 9 4 = − 65 4 . Using the angle sum formula for tangent, we have tan ( θ + 4 π ) = 1 − tan θ tan 4 π tan θ + tan 4 π = 1 − ( − 65 4 ) ⋅ 1 − 65 4 + 1 = 1 + 65 4 1 − 65 4 = 65 + 4 65 − 4 = ( 65 + 4 ) ( 65 − 4 ) ( 65 − 4 ) ( 65 − 4 ) = 65 − 16 65 − 8 65 + 16 = 49 81 − 8 65 .
Final Answer Therefore, the exact values are: (a) cos θ = − 9 65 (b) sin ( θ + 6 π ) = 18 4 3 − 65 (c) cos ( θ − 3 π ) = 18 4 3 − 65 (d) tan ( θ + 4 π ) = 49 81 − 8 65
Examples
Trigonometric functions and their identities are used extensively in physics, engineering, and navigation. For example, in physics, they are used to describe the motion of a pendulum or the oscillations of a spring. In engineering, they are used to analyze the stresses and strains in structures. In navigation, they are used to determine the position and direction of a ship or aircraft. Understanding how to manipulate trigonometric expressions and find exact values is crucial for solving problems in these fields. For instance, calculating the trajectory of a projectile involves trigonometric functions and their identities to accurately predict its path and landing point, considering factors like launch angle and initial velocity. Similarly, in electrical engineering, analyzing alternating current (AC) circuits relies heavily on trigonometric functions to model voltage and current waveforms.
The exact values are: (a) cos θ = − 9 65 , (b) sin ( θ + 6 π ) = 18 4 3 − 65 , (c) cos ( θ − 3 π ) = 18 4 3 − 65 , (d) tan ( θ + 4 π ) = 49 81 − 8 65 .
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