Question

based on the diagram, pick the two choices below that represent the expression \\(\cos(58)^\circ\\). (note: side lengths are rounded for simplicity, so the expressions may only be approximately equal.)
draw
triangle diagram with hypotenuse 53, one leg 28, and angle 58°
show your work here
hint: to add trig functions, type sin, cos, tan, ...
options:
\\(\frac{53}{28}\\)
\\(\sin(58^\circ)\\)
\\(\sin(32^\circ)\\)
\\(\cos(148^\circ)\\)
\\(\frac{28}{53}\\)

Explanation:

Step1: Recall cosine in right triangle

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. For $58^\circ$, adjacent side is $28$, hypotenuse is $53$. So $\cos(58^\circ)=\frac{28}{53}$.

Step2: Use co - function identity

We know that $\cos(\theta)=\sin(90^\circ - \theta)$. For $\theta = 58^\circ$, $90^\circ-58^\circ = 32^\circ$, so $\cos(58^\circ)=\sin(32^\circ)$.

Step3: Analyze other options

• $\frac{53}{28}$: This is $\frac{\text{hypotenuse}}{\text{adjacent}}$, not $\cos(58^\circ)$.
• $\sin(58^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}$, different from $\cos(58^\circ)$.
• $\cos(148^\circ)=\cos(180^\circ - 32^\circ)=-\cos(32^\circ)

eq\cos(58^\circ)$.

Answer:

$\sin(32^\circ)$, $\frac{28}{53}$