Question

solving problems with right triangles: mastery test
match each sine or cosine value to its equivalent measure.
cos(137°) sin(33°) sin(106°) sin(123°) cos(58°) cos(47°)
sin(16°)
sin(32°)
sin(43°)
cos(74°)
cos(57°)
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Explanation:

We use the co - function identities and the angle - addition/subtraction formulas for sine and cosine. Recall that $\sin(A)=\cos(90^{\circ}-A)$ and $\cos(A)=\sin(90^{\circ}-A)$, also $\sin(180^{\circ}-\theta)=\sin(\theta)$ and $\cos(180^{\circ}-\theta)=-\cos(\theta)$, but we can also use the co - function identity in a more general way for related angles.

Step 1: Match $\sin(32^{\circ})$

We know that $\cos(58^{\circ})=\cos(90^{\circ} - 32^{\circ})=\sin(32^{\circ})$ (by the co - function identity $\cos(x)=\sin(90^{\circ}-x)$).

Step 2: Match $\sin(43^{\circ})$

We know that $\cos(47^{\circ})=\cos(90^{\circ}-43^{\circ})=\sin(43^{\circ})$ (by the co - function identity $\cos(x)=\sin(90^{\circ}-x)$).

Step 3: Match $\cos(74^{\circ})$

We know that $\sin(16^{\circ})=\sin(90^{\circ}-74^{\circ})=\cos(74^{\circ})$ (by the co - function identity $\sin(x)=\cos(90^{\circ}-x)$).

Step 4: Match $\cos(57^{\circ})$

We know that $\sin(33^{\circ})=\sin(90^{\circ}-57^{\circ})=\cos(57^{\circ})$ (by the co - function identity $\sin(x)=\cos(90^{\circ}-x)$).

We can also check the other angles:

• $\cos(137^{\circ})=\cos(180^{\circ}-43^{\circ})=-\cos(43^{\circ})$, and $\sin(123^{\circ})=\sin(180^{\circ}-57^{\circ})=\sin(57^{\circ})$, $\sin(106^{\circ})=\sin(180^{\circ}-74^{\circ})=\sin(74^{\circ})$

Answer:

• $\sin(32^{\circ})\longleftrightarrow\cos(58^{\circ})$
• $\sin(43^{\circ})\longleftrightarrow\cos(47^{\circ})$
• $\cos(74^{\circ})\longleftrightarrow\sin(16^{\circ})$
• $\cos(57^{\circ})\longleftrightarrow\sin(33^{\circ})$