Select the correct answer.
For what value of $x$ is $\sin \left(x^{\circ}\right)=\cos \left(62^{\circ}\right)$, where $0^{\circ}
A. 31
B. 56
C. 14
D. 28
Answer (1)
Use the identity sin ( θ ) = cos ( 9 0 ∘ − θ ) to rewrite the equation.
Set 9 0 ∘ − x ∘ = 6 2 ∘ .
Solve for x .
The value of x is 28 .
Explanation
Analyze the problem We are given the equation sin ( x ∘ ) = cos ( 6 2 ∘ ) and we need to find the value of x such that 0 ∘ < x < 9 0 ∘ .
Apply trigonometric identity We know the trigonometric identity sin ( θ ) = cos ( 9 0 ∘ − θ ) . Using this identity, we can rewrite the given equation as: cos ( 9 0 ∘ − x ∘ ) = cos ( 6 2 ∘ )
Solve for x Since 0 ∘ < x < 9 0 ∘ , we have 0 ∘ < 9 0 ∘ − x < 9 0 ∘ . Therefore, we can equate the angles: 9 0 ∘ − x ∘ = 6 2 ∘ Now, we solve for x :
x = 90 − 62 x = 28
Final Answer Thus, the value of x is 28.
Examples
Understanding trigonometric relationships like sin ( x ) = cos ( 90 − x ) is useful in various fields. For example, in physics, when analyzing projectile motion, you often need to decompose velocity vectors into horizontal and vertical components. If you know the angle of projection, you can use these trigonometric identities to easily find the relationship between the components. Similarly, in engineering, when designing structures, you need to calculate forces at different angles, and these identities become essential tools for simplifying calculations.
