Finding unknown sides

Choosing the right trigonometric ratio and solving for an unknown side in the numerator or the denominator.

Method

Example 1●

Find $x$: the hypotenuse is $10$, the angle is $30^\circ$, and $x$ is opposite the angle.

$\sin 30^\circ = \dfrac{x}{10}$

$x = 10\sin 30^\circ = 5$

Example 2●●

Find $x$: the angle is $40^\circ$, the adjacent side is $12$, and $x$ is opposite.

$x = 12\tan 40^\circ \approx 10.07$

Example 3●●●

Find $x$: the angle is $35^\circ$, the opposite side is $7$, and $x$ is the hypotenuse.

$\sin 35^\circ = \dfrac{7}{x}$

$x = \dfrac{7}{\sin 35^\circ} \approx 12.20$

Flashcards

Unknown on top

Multiply: $x = 10\sin 30^\circ$

Unknown on the bottom

Divide: $x = \dfrac{7}{\sin 35^\circ}$

Calculator check

$\sin 30 = 0.5$ in degree mode.

Practice

Hypotenuse $10$, angle $30^\circ$, $x$ opposite. Find $x$.
Angle $40^\circ$, adjacent $12$, $x$ opposite. Find $x$ (2 d.p.).
Angle $35^\circ$, opposite $7$, $x$ the hypotenuse. Find $x$ (2 d.p.).
Hypotenuse $8$, angle $60^\circ$, $x$ adjacent. Find $x$.
Angle $50^\circ$, opposite $15$, $x$ adjacent. Find $x$ (2 d.p.).
Hypotenuse $20$, angle $25^\circ$, $x$ opposite. Find $x$ (2 d.p.).
Angle $42^\circ$, adjacent $6$, $x$ the hypotenuse. Find $x$ (2 d.p.).
The angle and the adjacent side are known; you want the opposite side. Which ratio?