Trigonometry (C) Word Problems (with Clues) worksheet featuring five geometry problems with diagrams and instructions.

Here are the step-by-step solutions for each problem. Remember to round your answers to 3 significant figures as requested.

1) Calculate the length of BC

* Identify the sides: We are looking at angle $C$ ($64^\circ$).
* Side $AB$ ($7\text{ cm}$) is Opposite to the angle.
* Side $BC$ is Adjacent to the angle.
* Choose the ratio: We have Opposite and need Adjacent, so we use Tangent ($\tan = \frac{\text{Opp}}{\text{Adj}}$).
* Set up the equation:
$$ \tan(64^\circ) = \frac{7}{BC} $$
* Rearrange to solve for BC:
$$ BC = \frac{7}{\tan(64^\circ)} $$
* Calculate:
$$ BC = \frac{7}{2.0503...} \approx 3.4141... $$
* Round to 3 s.f.: $3.41\text{ cm}$

2) Calculate the size of the other two angles

The hypotenuse is the longest side ($13\text{ cm}$). Let's find the angle opposite the $5\text{ cm}$ side first (let's call it $\theta$).
* Identify the sides relative to $\theta$:
* Opposite = $5\text{ cm}$
* Hypotenuse = $13\text{ cm}$
* Choose the ratio: We have Opposite and Hypotenuse, so we use Sine ($\sin = \frac{\text{Opp}}{\text{Hyp}}$).
* Set up the equation:
$$ \sin(\theta) = \frac{5}{13} $$
* Solve for $\theta$:
$$ \theta = \sin^{-1}\left(\frac{5}{13}\right) $$
$$ \theta \approx 22.619...^\circ $$
* Round to 3 s.f.: $22.6^\circ$
* Find the last angle: The angles in a triangle add up to $180^\circ$. Since one is $90^\circ$, the other two add up to $90^\circ$.
$$ 90^\circ - 22.619...^\circ = 67.380...^\circ $$
* Round to 3 s.f.: $67.4^\circ$

3) Calculate the size of the smallest angle

First, we must find the length of the missing side.
* Find the third side: Perimeter is $24\text{ cm}$. Known sides are $10\text{ cm}$ and $8\text{ cm}$.
$$ \text{Third side} = 24 - 10 - 8 = 6\text{ cm} $$
The sides are $6$, $8$, and $10$. (Check: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$, 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 $6\text{ cm}$. Let's call this angle $\alpha$.
* Identify sides relative to $\alpha$:
* Opposite = $6\text{ cm}$
* Hypotenuse = $10\text{ cm}$
* Choose the ratio: Use Sine.
$$ \sin(\alpha) = \frac{6}{10} = 0.6 $$
* Solve for $\alpha$:
$$ \alpha = \sin^{-1}(0.6) $$
$$ \alpha \approx 36.869...^\circ $$
* Round to 3 s.f.: $36.9^\circ$

4) What angle does the diagonal make with the longest side?

* Identify the triangle: The diagonal cuts the rectangle into two right triangles. The sides are $14\text{ cm}$ (longest side) and $8\text{ cm}$.
* Identify the angle: We want the angle touching the $14\text{ cm}$ side. Let's call it $\beta$.
* Identify sides relative to $\beta$:
* Opposite = $8\text{ cm}$
* Adjacent = $14\text{ cm}$
* Choose the ratio: Use Tangent.
$$ \tan(\beta) = \frac{8}{14} $$
* Solve for $\beta$:
$$ \beta = \tan^{-1}\left(\frac{8}{14}\right) $$
$$ \beta \approx 29.744...^\circ $$
* Round to 3 s.f.: $29.7^\circ$

5) How far up the wall does the ladder reach?

* Identify the sides:
* The ladder is the Hypotenuse ($5\text{ m}$).
* The height up the wall is Opposite to the $82^\circ$ angle.
* Choose the ratio: We have Hypotenuse and need Opposite, so use Sine.
$$ \sin(82^\circ) = \frac{\text{Height}}{5} $$
* Rearrange to solve for Height:
$$ \text{Height} = 5 \times \sin(82^\circ) $$
* Calculate:
$$ \text{Height} = 5 \times 0.9902... \approx 4.9513... $$
* Round to 3 s.f.: $4.95\text{ m}$

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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