Trigonometic Ratios (SOH CAH TOA): Word Problems (With Clues ... - Free Printable

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 ($\tan$).
$$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $$
* Substitute values:
$$ \tan(64^\circ) = \frac{7}{BC} $$
* 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 triangle has sides $5$, $12$, and $13$. Since $13$ is the longest side, it is the hypotenuse. Let's find the angle opposite the side of length $5$ (let's call it angle $A$) and the angle opposite the side of length $12$ (angle $B$).

* Find Angle A (opposite 5):
* Use Sine: $\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13}$
* $A = \sin^{-1}\left(\frac{5}{13}\right)$
* $A \approx 22.619...^\circ$
* Round to 3 s.f.: $22.6^\circ$

* Find Angle B (opposite 12):
* Use Cosine: $\cos(B) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{5}{13}$ (Side 5 is adjacent to angle B)
* Alternatively, use Sine: $\sin(B) = \frac{12}{13}$
* $B = \sin^{-1}\left(\frac{12}{13}\right)$
* $B \approx 67.380...^\circ$
* Round to 3 s.f.: $67.4^\circ$

*(Check: $22.6 + 67.4 + 90 = 180$. The angles add up correctly.)*

3) Calculate the size of the smallest angle.

* Find the missing side:
* Perimeter = $24 \text{ cm}$. Known sides are $10 \text{ cm}$ and $8 \text{ cm}$.
* Third side = $24 - 10 - 8 = 6 \text{ cm}$.
* The sides are $6$, $8$, and $10$.
* Identify the right angle:
* Check Pythagoras: $6^2 + 8^2 = 36 + 64 = 100$. $\sqrt{100} = 10$.
* So, $10$ is the hypotenuse. The right angle is opposite the side of length $10$.
* Find the smallest angle:
* The smallest angle is always opposite the shortest side. The shortest side is $6 \text{ cm}$.
* Let's call this angle $\theta$.
* $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{6}{10} = 0.6$
* $\theta = \sin^{-1}(0.6)$
* $\theta \approx 36.869...^\circ$
* Round to 3 s.f.: $36.9^\circ$

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

* Identify sides: Length = $14 \text{ cm}$ (Longest), Width = $8 \text{ cm}$.
* Visualize the triangle: The diagonal cuts the rectangle into two right-angled triangles. We want the angle between the diagonal and the side of length $14$.
* Identify relative to the angle:
* Side $14$ is Adjacent.
* Side $8$ is Opposite.
* Choose the ratio: Tangent ($\tan$).
$$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{8}{14} $$
* Calculate:
$$ \theta = \tan^{-1}\left(\frac{8}{14}\right) $$
$$ \theta = \tan^{-1}(0.5714...) $$
$$ \theta \approx 29.744...^\circ $$
* Round to 3 s.f.: $29.7^\circ$

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

* Identify the sides:
* Ladder length = $5 \text{ m}$ (Hypotenuse).
* Angle with ground = $82^\circ$.
* Height up the wall is Opposite to the $82^\circ$ angle.
* Choose the ratio: Sine ($\sin$).
$$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
* Substitute values:
$$ \sin(82^\circ) = \frac{\text{Height}}{5} $$
* Rearrange:
$$ \text{Height} = 5 \times \sin(82^\circ) $$
* Calculate:
$$ \text{Height} = 5 \times 0.9902... $$
$$ \text{Height} \approx 4.9513... \text{ m} $$
* Round to 3 s.f.: $4.95 \text{ m}$

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