Given that $\sin \theta=\frac{1}{7}$ and that the terminal side is in quadrant II, complete parts a through c.
a) Give an exact answer for the other function values for $\theta$.
$\cos \theta=\square$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answer (1)
Use the Pythagorean identity sin 2 θ + cos 2 θ = 1 .
Substitute sin θ = 7 1 into the identity and solve for cos 2 θ .
Take the square root to find cos θ , considering the quadrant to determine the sign.
Since θ is in quadrant II, cos θ is negative, so − 7 4 3 .
Explanation
Problem Analysis We are given that sin θ = 7 1 and that the terminal side of θ is in quadrant II. We need to find the exact value of cos θ .
Apply Pythagorean Identity We can use the Pythagorean identity sin 2 θ + cos 2 θ = 1 to find cos θ . Substituting the given value of sin θ into the identity, we get: ( 7 1 ) 2 + cos 2 θ = 1
Isolate cos^2(theta) Solving for cos 2 θ , we have: cos 2 θ = 1 − ( 7 1 ) 2 = 1 − 49 1 = 49 49 − 49 1 = 49 48
Take Square Root Taking the square root of both sides, we get: cos θ = ± 49 48 = ± 49 48 = ± 7 16 × 3 = ± 7 4 3
Determine Sign Since the terminal side of θ is in quadrant II, cos θ is negative. Therefore, cos θ = − 7 4 3
Examples
Understanding trigonometric functions and their relationships, like the Pythagorean identity, is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum, knowing the sine of the angle allows you to determine the cosine, which is essential for calculating the pendulum's potential and kinetic energy. This ensures accurate predictions of the pendulum's behavior.
