Write the function value in terms of the cofunction of a complementary angle.
$\cos 141^{\circ} 51^{\prime}=$
Answer (1)
Find the reference angle for 14 1 ∘ 5 1 ′ , which is 3 8 ∘ 9 ′ .
Determine the sign of cos 14 1 ∘ 5 1 ′ in the second quadrant, which is negative.
Use the cofunction identity cos x = sin ( 9 0 ∘ − x ) to express cos 3 8 ∘ 9 ′ as sin 5 1 ∘ 9 ′ .
Substitute back to get the final answer: − sin 5 1 ∘ 9 ′ .
Explanation
Problem Analysis We are asked to express cos 14 1 ∘ 5 1 ′ in terms of the cofunction of a complementary angle. This means we need to find an angle θ and a cofunction such that cos 14 1 ∘ 5 1 ′ can be written as either sin ( θ ) , − sin ( θ ) , cos ( θ ) , − cos ( θ ) , tan ( θ ) , − tan ( θ ) , cot ( θ ) , or − cot ( θ ) .
Find the Reference Angle First, let's find the reference angle for 14 1 ∘ 5 1 ′ . Since 14 1 ∘ 5 1 ′ is in the second quadrant, the reference angle is calculated as:
18 0 ∘ − 14 1 ∘ 5 1 ′ = 3 8 ∘ 9 ′
Determine the Sign Since cosine is negative in the second quadrant, we have:
cos 14 1 ∘ 5 1 ′ = − cos 3 8 ∘ 9 ′
Use the Cofunction Identity Now, we use the identity cos x = sin ( 9 0 ∘ − x ) . Therefore:
cos 3 8 ∘ 9 ′ = sin ( 9 0 ∘ − 3 8 ∘ 9 ′ )
Calculating the complementary angle:
9 0 ∘ − 3 8 ∘ 9 ′ = 5 1 ∘ 5 1 ′
Substitute Back So, we have:
cos 3 8 ∘ 9 ′ = sin 5 1 ∘ 9 ′
Substituting this back into our original equation:
cos 14 1 ∘ 5 1 ′ = − sin 5 1 ∘ 9 ′
Final Answer Therefore, the function value in terms of the cofunction of a complementary angle is:
cos 14 1 ∘ 5 1 ′ = − sin 5 1 ∘ 9 ′
Examples
Understanding cofunctions and complementary angles is useful in various fields, such as physics and engineering, where angles and trigonometric functions are used to analyze forces, motion, and wave phenomena. For example, when analyzing projectile motion, you might need to decompose the initial velocity into horizontal and vertical components using sine and cosine. Knowing the relationship between these functions and their complementary angles can simplify calculations and provide a deeper understanding of the problem.
