Question

which expressions represent the length of y? please select two.
draw
show your work here
hint: to add trig functions, type sin, cos, tan, ...
$18 \cdot \cos(65^\circ)$ $\frac{18}{\sin(25^\circ)}$
$\frac{18}{\cos(65^\circ)}$ $18 \cdot \cos(25^\circ)$
$18 \cdot \sin(25^\circ)$

Explanation:

Step1: Analyze the right triangle

We have a right triangle with one angle \( 65^\circ \), the adjacent side to \( 65^\circ \) is \( 18 \), and \( y \) is the hypotenuse. Also, the other non - right angle is \( 90^\circ - 65^\circ=25^\circ \).

Step2: Use trigonometric ratios

For the \( 65^\circ \) angle:

The cosine of an angle in a right triangle is defined as \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \theta = 65^\circ \), \( \cos(65^\circ)=\frac{18}{y} \), so we can solve for \( y \):
\[
y=\frac{18}{\cos(65^\circ)}
\]

For the \( 25^\circ \) angle:

The sine of an angle in a right triangle is defined as \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \theta = 25^\circ \), the opposite side to \( 25^\circ \) is \( 18 \), and \( \sin(25^\circ)=\frac{18}{y} \), so we can solve for \( y \):
\[
y = \frac{18}{\sin(25^\circ)}
\]
Also, we can use the cosine of \( 25^\circ \) (since \( \cos(25^\circ)=\sin(65^\circ) \)) or sine of \( 65^\circ \) to check other expressions. But from the given options, we analyze each option:

• Option \( 18\cdot\cos(65^\circ) \): \( \cos(65^\circ)=\frac{18}{y}\Rightarrow18\cdot\cos(65^\circ)=\frac{18\times18}{y}

eq y \), so this is wrong.

• Option \( \frac{18}{\sin(25^\circ)} \): As we derived above, this is correct.
• Option \( \frac{18}{\cos(65^\circ)} \): As we derived above, this is correct.
• Option \( 18\cdot\cos(25^\circ) \): \( \cos(25^\circ)=\frac{y}{?} \) (not relevant to the relation for \( y \) with adjacent side 18), so this is wrong.
• Option \( 18\cdot\sin(25^\circ) \): \( \sin(25^\circ)=\frac{18}{y}\Rightarrow18\cdot\sin(25^\circ)=\frac{18\times18}{y}

eq y \), so this is wrong.

Answer:

The two expressions that represent the length of \( y \) are \( \frac{18}{\cos(65^\circ)} \) and \( \frac{18}{\sin(25^\circ)} \) (or in the boxed form as per the options, the correct options are the ones with \( \frac{18}{\cos(65^\circ)} \) and \( \frac{18}{\sin(25^\circ)} \))