Right Triangle Trig Missing Sides and ... - 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 and angles.

Remember SOH CAH TOA:
* Sin = Opposite / Hypotenuse
* Cos = Adjacent / Hypotenuse
* Tan = Opposite / Adjacent

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Part 1: Find the measure of each angle indicated (Round to the nearest tenth)

1) Find angle $A$ ($\theta$)
* Identify sides: Relative to angle $A$, side $BC$ (12) is Opposite, and side $AB$ (13) is the Hypotenuse.
* Choose ratio: Sine ($\sin = \text{Opp}/\text{Hyp}$).
* Equation: $\sin(A) = \frac{12}{13}$
* Calculate: $A = \sin^{-1}(\frac{12}{13}) \approx 67.38^\circ$
* Round: $67.4^\circ$

2) Find angle $A$ ($\theta$)
* Identify sides: Relative to angle $A$, side $BC$ (4) is Opposite, and side $AB$ (13) is the Hypotenuse.
* Choose ratio: Sine.
* Equation: $\sin(A) = \frac{4}{13}$
* Calculate: $A = \sin^{-1}(\frac{4}{13}) \approx 17.92^\circ$
* Round: $17.9^\circ$

3) Find angle $A$ ($\theta$)
* Identify sides: Relative to angle $A$, side $BC$ (9) is Opposite, and side $AC$ (6) is Adjacent.
* Choose ratio: Tangent ($\tan = \text{Opp}/\text{Adj}$).
* Equation: $\tan(A) = \frac{9}{6}$
* Calculate: $A = \tan^{-1}(1.5) \approx 56.31^\circ$
* Round: $56.3^\circ$

4) Find angle $B$ ($\theta$)
* Identify sides: Relative to angle $B$, side $AC$ (10) is Opposite, and side $BC$ (11.9) is Adjacent.
* Choose ratio: Tangent.
* Equation: $\tan(B) = \frac{10}{11.9}$
* Calculate: $B = \tan^{-1}(\frac{10}{11.9}) \approx 40.03^\circ$
* Round: $40.0^\circ$

5) Find angle $B$ ($\theta$)
* Identify sides: Relative to angle $B$, side $AC$ (7.7) is Opposite, and side $AB$ (14) is the Hypotenuse.
* Choose ratio: Sine.
* Equation: $\sin(B) = \frac{7.7}{14}$
* Calculate: $B = \sin^{-1}(0.55) \approx 33.37^\circ$
* Round: $33.4^\circ$

6) Find angle $B$ ($\theta$)
* Identify sides: Relative to angle $B$, side $AC$ (4) is Adjacent, and side $AB$ (5) is the Hypotenuse.
* Choose ratio: Cosine ($\cos = \text{Adj}/\text{Hyp}$).
* Equation: $\cos(B) = \frac{4}{5}$
* Calculate: $B = \cos^{-1}(0.8) \approx 36.87^\circ$
* Round: $36.9^\circ$

7) Find angle $B$ ($\theta$)
* Identify sides: Relative to angle $B$, side $AC$ (4.4) is Opposite, and side $AB$ (11) is the Hypotenuse.
* Choose ratio: Sine.
* Equation: $\sin(B) = \frac{4.4}{11}$
* Calculate: $B = \sin^{-1}(0.4) \approx 23.58^\circ$
* Round: $23.6^\circ$

8) Find angle $B$ ($\theta$)
* Identify sides: Relative to angle $B$, side $AC$ (3) is Opposite, and side $BC$ (3) is Adjacent.
* Choose ratio: Tangent.
* Equation: $\tan(B) = \frac{3}{3} = 1$
* Calculate: $B = \tan^{-1}(1) = 45^\circ$
* Round: $45.0^\circ$

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Part 2: Find the measure of each side indicated (Round to the nearest tenth)

9) Find side $x$ ($AB$)
* Given: Angle $B = 57^\circ$, Side $BC$ (Adjacent) = 11. Find Hypotenuse $x$.
* Choose ratio: Cosine.
* Equation: $\cos(57^\circ) = \frac{11}{x}$
* Rearrange: $x = \frac{11}{\cos(57^\circ)}$
* Calculate: $x \approx \frac{11}{0.5446} \approx 20.198$
* Round: $20.2$

10) Find side $x$ ($BC$)
* Given: Angle $A = 36^\circ$, Side $AC$ (Adjacent) = 13. Find Opposite side $x$.
* Choose ratio: Tangent.
* Equation: $\tan(36^\circ) = \frac{x}{13}$
* Rearrange: $x = 13 \cdot \tan(36^\circ)$
* Calculate: $x \approx 13 \cdot 0.7265 \approx 9.445$
* Round: $9.4$

11) Find side $x$ ($CB$)
* Given: Angle $A = 50.1^\circ$, Side $AC$ (Adjacent) = 5. Find Opposite side $x$.
* Choose ratio: Tangent.
* Equation: $\tan(50.1^\circ) = \frac{x}{5}$
* Rearrange: $x = 5 \cdot \tan(50.1^\circ)$
* Calculate: $x \approx 5 \cdot 1.196 \approx 5.98$
* Round: $6.0$

12) Find side $x$ ($BC$)
* Given: Angle $B = 60^\circ$, Side $AB$ (Hypotenuse) = 11. Find Adjacent side $x$.
* Choose ratio: Cosine.
* Equation: $\cos(60^\circ) = \frac{x}{11}$
* Rearrange: $x = 11 \cdot \cos(60^\circ)$
* Calculate: $x = 11 \cdot 0.5 = 5.5$
* Round: $5.5$

Final Answer:
1) 67.4°
2) 17.9°
3) 56.3°
4) 40.0°
5) 33.4°
6) 36.9°
7) 23.6°
8) 45.0°
9) 20.2
10) 9.4
11) 6.0
12) 5.5