Right Triangle Trig Finding Missing Sides And Angles Worksheet Answers - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use trigonometric ratios (Sine, Cosine, Tangent) to find the missing sides.

Reminder of Ratios (SOH CAH TOA):
* Sine = Opposite / Hypotenuse
* Cosine = Adjacent / Hypotenuse
* Tangent = Opposite / Adjacent

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1) Find f
* Identify: Angle is $55^\circ$. Side $f$ is Opposite. The side labeled 40 is the Hypotenuse (longest side).
* Ratio: Sine ($\sin = \text{Opp} / \text{Hyp}$).
* Equation: $\sin(55^\circ) = f / 40$
* Solve: $f = 40 \times \sin(55^\circ)$
* Calculate: $40 \times 0.81915... \approx 32.766$
* Round: $32.8$

2) Find y
* Identify: Angle is $81^\circ$. Side $y$ is Opposite. The side labeled 24 is Adjacent.
* Ratio: Tangent ($\tan = \text{Opp} / \text{Adj}$).
* Equation: $\tan(81^\circ) = y / 24$
* Solve: $y = 24 \times \tan(81^\circ)$
* Calculate: $24 \times 6.3137... \approx 151.529$
* Round: $151.5$

3) Find o
* Identify: Angle is $37^\circ$. Side $o$ is Opposite. The side labeled 12 is Adjacent.
* Ratio: Tangent ($\tan = \text{Opp} / \text{Adj}$).
* Equation: $\tan(37^\circ) = o / 12$
* Solve: $o = 12 \times \tan(37^\circ)$
* Calculate: $12 \times 0.75355... \approx 9.042$
* Round: $9.0$

4) Find o
* Identify: Angle is $35^\circ$. Side 24 is Opposite. Side $o$ is Adjacent.
* Ratio: Tangent ($\tan = \text{Opp} / \text{Adj}$).
* Equation: $\tan(35^\circ) = 24 / o$
* Solve: Multiply by $o$, then divide by $\tan(35^\circ)$. So, $o = 24 / \tan(35^\circ)$.
* Calculate: $24 / 0.7002... \approx 34.275$
* Round: $34.3$

5) Find j
* Identify: Angle is $36^\circ$. Side $j$ is Adjacent. The side labeled 45 is the Hypotenuse.
* Ratio: Cosine ($\cos = \text{Adj} / \text{Hyp}$).
* Equation: $\cos(36^\circ) = j / 45$
* Solve: $j = 45 \times \cos(36^\circ)$
* Calculate: $45 \times 0.8090... \approx 36.405$
* Round: $36.4$

6) Find v
* Identify: Angle is $34^\circ$. Side $v$ is Opposite. The side labeled 37 is the Hypotenuse.
* Ratio: Sine ($\sin = \text{Opp} / \text{Hyp}$).
* Equation: $\sin(34^\circ) = v / 37$
* Solve: $v = 37 \times \sin(34^\circ)$
* Calculate: $37 \times 0.55919... \approx 20.69$
* Round: $20.7$

7) Find v
* Identify: Angle is $38^\circ$. Side $v$ is Adjacent. The side labeled 45 is the Hypotenuse.
* Ratio: Cosine ($\cos = \text{Adj} / \text{Hyp}$).
* Equation: $\cos(38^\circ) = v / 45$
* Solve: $v = 45 \times \cos(38^\circ)$
* Calculate: $45 \times 0.7880... \approx 35.46$
* Round: $35.5$

8) Find f
* Identify: Angle is $48^\circ$. Side 20 is Opposite. Side $f$ is Adjacent.
* Ratio: Tangent ($\tan = \text{Opp} / \text{Adj}$).
* Equation: $\tan(48^\circ) = 20 / f$
* Solve: $f = 20 / \tan(48^\circ)$
* Calculate: $20 / 1.1106... \approx 18.008$
* Round: $18.0$

Final Answer:
1) f = 32.8
2) y = 151.5
3) o = 9.0
4) o = 34.3
5) j = 36.4
6) v = 20.7
7) v = 35.5
8) f = 18.0