Chapter 12: Worksheet Honors Geometry Name: Trig Problem Solving ... - Free Printable

Here are the step-by-step solutions for the problems on your worksheet.

Rounding Rules:
* Lengths: Round to the nearest hundredth (2 decimal places).
* Angles: Round to the nearest tenth (1 decimal place).

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Problems 1–6: Find the missing side or angle

1. Find $w$
* Identify sides: Relative to the $20^\circ$ angle, $w$ is the Opposite side and $24$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Equation: $\sin(20^\circ) = \frac{w}{24}$
* Solve: $w = 24 \cdot \sin(20^\circ)$
* Calculation: $w \approx 24 \cdot 0.3420 = 8.208...$
* Round: $8.21$ cm

2. Find $x$
* Identify sides: Relative to the $32^\circ$ angle, $x$ is the Adjacent side and $17$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Equation: $\cos(32^\circ) = \frac{x}{17}$
* Solve: $x = 17 \cdot \cos(32^\circ)$
* Calculation: $x \approx 17 \cdot 0.8480 = 14.416...$
* Round: $14.42$ cm

3. Find $y$
* Identify sides: Relative to the $50^\circ$ angle, $20$ is the Opposite side and $y$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Equation: $\tan(50^\circ) = \frac{20}{y}$
* Solve: $y = \frac{20}{\tan(50^\circ)}$
* Calculation: $y \approx \frac{20}{1.1917} = 16.782...$
* Round: $16.78$ cm

4. Find $q$
* Identify sides: Relative to the $47^\circ$ angle, $12$ is the Adjacent side and $q$ is the Hypotenuse.
* Formula: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Equation: $\cos(47^\circ) = \frac{12}{q}$
* Solve: $q = \frac{12}{\cos(47^\circ)}$
* Calculation: $q \approx \frac{12}{0.6820} = 17.595...$
* Round: $17.60$ cm

5. Find angle $g$
* Identify sides: Relative to angle $g$, $6$ is the Opposite side and $11$ is the Adjacent side.
* Formula: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
* Equation: $\tan(g) = \frac{6}{11}$
* Solve: $g = \tan^{-1}\left(\frac{6}{11}\right)$
* Calculation: $g \approx \tan^{-1}(0.5454) = 28.610...^\circ$
* Round: $28.6^\circ$

6. Find angle $d$
* Identify sides: Relative to angle $d$, $5$ is the Opposite side and $13$ is the Hypotenuse.
* Formula: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
* Equation: $\sin(d) = \frac{5}{13}$
* Solve: $d = \sin^{-1}\left(\frac{5}{13}\right)$
* Calculation: $d \approx \sin^{-1}(0.3846) = 22.619...^\circ$
* Round: $22.6^\circ$

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Problems 7 & 8: Find the indicated measures

7. Find $BC$ and $BD$

* Part a: Find $BC$
* Look at the large right triangle $\triangle ABC$. We know hypotenuse $AB = 30$ and angle $A = 25^\circ$.
* $BC$ is opposite to angle $A$.
* Equation: $\sin(25^\circ) = \frac{BC}{30}$
* Solve: $BC = 30 \cdot \sin(25^\circ)$
* Calculation: $BC \approx 30 \cdot 0.4226 = 12.678...$
* Round: $12.68$ in.

* Part b: Find $BD$
* Look at the smaller right triangle $\triangle BCD$. We just found leg $BC \approx 12.678$. We know angle $\angle BDC = 50^\circ$.
* $BD$ is the hypotenuse, and $BC$ is opposite the $50^\circ$ angle.
* Equation: $\sin(50^\circ) = \frac{BC}{BD}$
* Solve: $BD = \frac{BC}{\sin(50^\circ)}$
* Calculation: $BD \approx \frac{12.678}{0.7660} = 16.550...$
* Round: $16.55$ in.

8. Find $EF$, $EG$, and $FG$

* Part a: Find $EF$
* Look at right triangle $\triangle FEH$. We know leg $EH = 32$ and angle $\angle FHE = 48^\circ$.
* $EF$ is opposite the $48^\circ$ angle, and $EH$ is adjacent.
* Equation: $\tan(48^\circ) = \frac{EF}{32}$
* Solve: $EF = 32 \cdot \tan(48^\circ)$
* Calculation: $EF \approx 32 \cdot 1.1106 = 35.539...$
* Round: $35.54$ cm

* Part b: Find $EG$
* Look at the large right triangle $\triangle GEH$. We know leg $EH = 32$.
* The total angle at $H$ is $\angle GHE = 48^\circ + 35^\circ = 83^\circ$.
* $EG$ is opposite the $83^\circ$ angle.
* Equation: $\tan(83^\circ) = \frac{EG}{32}$
* Solve: $EG = 32 \cdot \tan(83^\circ)$
* Calculation: $EG \approx 32 \cdot 8.1443 = 260.619...$
* Round: $260.62$ cm

* Part c: Find $FG$
* $FG$ is the top segment of the vertical line. It equals the total height ($EG$) minus the bottom height ($EF$).
* Equation: $FG = EG - EF$
* Calculation: $FG \approx 260.62 - 35.54 = 225.08$
* Round: $225.08$ cm

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Final Answer:
1. $w = 8.21$ cm
2. $x = 14.42$ cm
3. $y = 16.78$ cm
4. $q = 17.60$ cm
5. $g = 28.6^\circ$
6. $d = 22.6^\circ$
7. a. $BC = 12.68$ in.
b. $BD = 16.55$ in.
8. a. $EF = 35.54$ cm
b. $EG = 260.62$ cm
c. $FG = 225.08$ cm