Trigonometry (C) Word Problems worksheet featuring five math problems related to right triangles and angles.

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

1) Calculate the length of BC.
* Identify the sides: We have a right triangle ABC with the right angle at B ($90^\circ$). We know angle C is $64^\circ$. The side opposite to angle C is AB ($7\text{ cm}$), and the side next to angle C (adjacent) is BC, which we need to find.
* Choose the ratio: Since we have the Opposite side and want the Adjacent side, we use Tangent ($\tan = \frac{\text{Opposite}}{\text{Adjacent}}$).
* Set up the equation: $\tan(64^\circ) = \frac{7}{BC}$.
* Solve: To get BC by itself, we rearrange the formula: $BC = \frac{7}{\tan(64^\circ)}$.
* Calculate: $BC \approx \frac{7}{2.0503} \approx 3.4141...$
* Round: To 3 significant figures, this is 3.41 cm.

2) Calculate the size of the other two angles.
* Identify the sides: The sides are 5, 12, and 13. Since 13 is the longest side, it is the hypotenuse. This means the right angle is opposite the side of length 13.
* Find Angle A: Let's find the angle opposite the side of length 5. We use Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* $\sin(A) = \frac{5}{13}$.
* $A = \sin^{-1}(\frac{5}{13}) \approx 22.619...^\circ$
* Rounded to 3 sig figs: 22.6°.
* Find Angle B: The angles in a triangle add up to $180^\circ$. Since one is $90^\circ$ and another is $22.6^\circ$:
* $B = 180 - 90 - 22.6 = 67.4^\circ$.
* (Check using Cosine: $\cos(B) = \frac{5}{13} \rightarrow B \approx 67.38^\circ$, which rounds to 67.4°).
* Answer: The angles are 22.6° and 67.4°.

3) Calculate the size of the smallest angle.
* Find the missing side: The perimeter is $24\text{ cm}$. Two sides are $10\text{ cm}$ and $8\text{ cm}$.
* Third side = $24 - 10 - 8 = 6\text{ cm}$.
* The sides are 6, 8, and 10. (Note: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$, so it is a right triangle with hypotenuse 10).
* Identify the smallest angle: The smallest angle is always opposite the shortest side. The shortest side is 6.
* Calculate: Use Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* $\sin(\theta) = \frac{6}{10} = 0.6$.
* $\theta = \sin^{-1}(0.6) \approx 36.869...^\circ$
* Round: To 3 significant figures, this is 36.9°.

4) What angle does the diagonal make with the longest side?
* Visualize: A rectangle cut by a diagonal forms a right triangle. The legs are $14\text{ cm}$ and $8\text{ cm}$. The "longest side" of the rectangle is the base, $14\text{ cm}$.
* Set up: We want the angle between the diagonal (hypotenuse) and the base ($14\text{ cm}$).
* Adjacent side = 14.
* Opposite side = 8.
* Choose ratio: Use Tangent ($\tan = \frac{\text{Opposite}}{\text{Adjacent}}$).
* $\tan(\theta) = \frac{8}{14}$.
* Calculate: $\theta = \tan^{-1}(\frac{8}{14}) \approx 29.744...^\circ$
* Round: To 3 significant figures, this is 29.7°.

5) How far up the wall does the ladder reach?
* Identify parts: The ladder is the hypotenuse ($5\text{ m}$). The angle with the ground is $82^\circ$. We want to find the height up the wall, which is the side opposite the angle.
* Choose ratio: Use Sine ($\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$).
* Set up: $\sin(82^\circ) = \frac{\text{Height}}{5}$.
* Solve: $\text{Height} = 5 \times \sin(82^\circ)$.
* Calculate: $\text{Height} \approx 5 \times 0.99026... \approx 4.9513...$
* Round: To 3 significant figures, this is 4.95 m.

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Final Answer:
1) 3.41 cm
2) 22.6° and 67.4°
3) 36.9°
4) 29.7°
5) 4.95 m