Given that [tex]$\sin \theta=\frac{21}{29}$[/tex], what is the value of [tex]$\cos \theta$[/tex], for [tex]$0^{\circ}\ \textless \ \theta\ \textless \ 90^{\circ}$[/tex] ?
A. [tex]$-\sqrt{\frac{20}{29}}$[/tex]
B. [tex]$-\frac{20}{29}$[/tex]
C. [tex]$\frac{20}{29}$[/tex]
D. [tex]$\sqrt{\frac{20}{29}}$[/tex]
Answer (2)
Use the Pythagorean identity sin 2 θ + cos 2 θ = 1 .
Substitute the given sin θ = 29 21 into the identity.
Solve for cos 2 θ and then find cos θ by taking the square root.
Since 0 ∘ < θ < 9 0 ∘ , choose the positive value for cos θ : 29 20 .
Explanation
Problem Analysis We are given that sin θ = 29 21 and 0 ∘ < θ < 9 0 ∘ . We want to find the value of cos θ .
Using Pythagorean Identity We can use the Pythagorean identity, which states that sin 2 θ + cos 2 θ = 1 .
Substitution Substitute the given value of sin θ into the identity: ( 29 21 ) 2 + cos 2 θ = 1
Solving for cos^2(theta) Solve for cos 2 θ :
cos 2 θ = 1 − ( 29 21 ) 2 cos 2 θ = 1 − 841 441 cos 2 θ = 841 841 − 441 cos 2 θ = 841 400
Finding cos(theta) Take the square root of both sides to solve for cos θ :
cos θ = ± 841 400 cos θ = ± 29 20
Determining the sign of cos(theta) Since 0 ∘ < θ < 9 0 ∘ , θ is in the first quadrant, where cosine is positive. Therefore, we take the positive value: cos θ = 29 20
Final Answer The value of cos θ is 29 20 .
Examples
Understanding trigonometric relationships like sin θ and cos θ is crucial in fields like navigation and physics. For example, when calculating the trajectory of a projectile, knowing the launch angle θ and initial velocity allows us to determine the horizontal and vertical components of the velocity using sin θ and cos θ . These components are essential for predicting the range and maximum height of the projectile. Similarly, in navigation, sailors use trigonometric functions to calculate distances and bearings, ensuring they stay on course.
The value of cos θ given that sin θ = 29 21 is 29 20 . This is determined using the Pythagorean identity and finding the appropriate square root. Thus, the correct choice is C. 29 20 .
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