It is given that [tex]$\sin 50^{\circ}=0.766$[/tex] and [tex]$\cos 124^{\circ}=-0.559$[/tex] when corrected to three decimal places. Without using a calculator, find the value of [tex]$\sin 130^{\circ}+\cos 56^{\circ}$[/tex].

Use the identity sin ( 18 0 ∘ − x ) = sin x to find that sin 13 0 ∘ = sin 5 0 ∘ = 0.766 .
Use the identity cos ( 9 0 ∘ − x ) = sin x to express cos 5 6 ∘ as sin ( 9 0 ∘ − 5 6 ∘ ) = sin 3 4 ∘ .
Use the fact that cos 12 4 ∘ = − sin 3 4 ∘ to find that sin 3 4 ∘ = − cos 12 4 ∘ = 0.559 .
Calculate sin 13 0 ∘ + cos 5 6 ∘ = 0.766 + 0.559 = 1.325 ​ .

Explanation

Problem Analysis We are given that sin 5 0 ∘ = 0.766 and cos 12 4 ∘ = − 0.559 . We need to find the value of sin 13 0 ∘ + cos 5 6 ∘ without using a calculator.

Calculate sin 13 0 ∘ First, we can use the identity sin ( 18 0 ∘ − x ) = sin x to express sin 13 0 ∘ in terms of sin 5 0 ∘ .
sin 13 0 ∘ = sin ( 18 0 ∘ − 5 0 ∘ ) = sin 5 0 ∘ = 0.766

Express cos 5 6 ∘ in terms of sin 3 4 ∘ Next, we need to find cos 5 6 ∘ . We can use the identity cos ( 9 0 ∘ − x ) = sin x to express cos 5 6 ∘ as sin ( 9 0 ∘ − 5 6 ∘ ) = sin 3 4 ∘ .
So, cos 5 6 ∘ = sin 3 4 ∘ .

Find sin 6 2 ∘ Now, we need to find sin 3 4 ∘ . We can use the identity cos ( 2 x ) = 1 − 2 sin 2 ( x ) . We are given cos 12 4 ∘ = − 0.559 . Let 2 x = 12 4 ∘ , so x = 6 2 ∘ .
cos 12 4 ∘ = 1 − 2 sin 2 6 2 ∘ − 0.559 = 1 − 2 sin 2 6 2 ∘ 2 sin 2 6 2 ∘ = 1 + 0.559 = 1.559 sin 2 6 2 ∘ = 2 1.559 ​ = 0.7795 sin 6 2 ∘ = 0.7795 ​ ≈ 0.883

Find sin 3 4 ∘ We can use the identity sin ( 2 x ) = 2 sin x cos x . Let 2 x = 6 8 ∘ , so x = 3 4 ∘ .
We know that cos 12 4 ∘ = − 0.559 . Also, cos 12 4 ∘ = cos ( 9 0 ∘ + 3 4 ∘ ) = − sin 3 4 ∘ .
Therefore, sin 3 4 ∘ = − cos 12 4 ∘ = − ( − 0.559 ) = 0.559 .

Calculate the final value Now we can calculate sin 13 0 ∘ + cos 5 6 ∘ .
sin 13 0 ∘ + cos 5 6 ∘ = sin 5 0 ∘ + sin 3 4 ∘ = 0.766 + 0.559 = 1.325

Final Answer Therefore, the value of sin 13 0 ∘ + cos 5 6 ∘ is 1.325 .


Examples
Understanding trigonometric identities helps in various fields like physics and engineering. For example, when analyzing projectile motion, we often need to decompose the initial velocity into horizontal and vertical components using sine and cosine functions. Similarly, in electrical engineering, alternating current (AC) circuits are analyzed using sinusoidal functions, where understanding phase shifts and trigonometric relationships is crucial for circuit design and analysis.

Answered by GinnyAnswer | 2025-07-05

The value of sin 13 0 ∘ + cos 5 6 ∘ is 1.325 . This is derived using trigonometric identities to express both functions in terms of the values given. Specifically, we find sin 13 0 ∘ = 0.766 and cos 5 6 ∘ = 0.559 .
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Answered by Anonymous | 2025-07-12