To solve the missing side lengths in the given right triangles, we will use trigonometric ratios: sine, cosine, and tangent. Let's go through each part step by step.
---
Part (a)
Given:
- Hypotenuse = 8 cm
- Angle = 39°
- Missing side is adjacent to the 39° angle.
We use the cosine ratio:
\[
\cos(39^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
\cos(39^\circ) = \frac{x}{8}
\]
Solving for \( x \):
\[
x = 8 \cdot \cos(39^\circ)
\]
Using a calculator:
\[
\cos(39^\circ) \approx 0.7771
\]
\[
x \approx 8 \cdot 0.7771 \approx 6.2
\]
Answer for (a): \( x \approx 6.2 \) cm
---
Part (b)
Given:
- One leg = 10 cm
- Angle = 49°
- Missing side is the hypotenuse.
We use the sine ratio:
\[
\sin(49^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin(49^\circ) = \frac{10}{y}
\]
Solving for \( y \):
\[
y = \frac{10}{\sin(49^\circ)}
\]
Using a calculator:
\[
\sin(49^\circ) \approx 0.7547
\]
\[
y \approx \frac{10}{0.7547} \approx 13.3
\]
Answer for (b): \( y \approx 13.3 \) cm
---
Part (c)
Given:
- One leg = 6 cm
- Angle = 57°
- Missing side is the opposite side.
We use the tangent ratio:
\[
\tan(57^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(57^\circ) = \frac{z}{6}
\]
Solving for \( z \):
\[
z = 6 \cdot \tan(57^\circ)
\]
Using a calculator:
\[
\tan(57^\circ) \approx 1.5399
\]
\[
z \approx 6 \cdot 1.5399 \approx 9.2
\]
Answer for (c): \( z \approx 9.2 \) cm
---
Part (d)
Given:
- One leg = 8 cm
- Angle = 35°
- Missing side is the opposite side.
We use the tangent ratio:
\[
\tan(35^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(35^\circ) = \frac{w}{8}
\]
Solving for \( w \):
\[
w = 8 \cdot \tan(35^\circ)
\]
Using a calculator:
\[
\tan(35^\circ) \approx 0.7002
\]
\[
w \approx 8 \cdot 0.7002 \approx 5.6
\]
Answer for (d): \( w \approx 5.6 \) cm
---
Part (e)
Given:
- One leg = 5 cm
- Angle = 42°
- Missing side is the hypotenuse.
We use the cosine ratio:
\[
\cos(42^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
\cos(42^\circ) = \frac{5}{v}
\]
Solving for \( v \):
\[
v = \frac{5}{\cos(42^\circ)}
\]
Using a calculator:
\[
\cos(42^\circ) \approx 0.7431
\]
\[
v \approx \frac{5}{0.7431} \approx 6.7
\]
Answer for (e): \( v \approx 6.7 \) cm
---
Part (f)
Given:
- One leg = 6 cm
- Angle = 28°
- Missing side is the opposite side.
We use the tangent ratio:
\[
\tan(28^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(28^\circ) = \frac{u}{6}
\]
Solving for \( u \):
\[
u = 6 \cdot \tan(28^\circ)
\]
Using a calculator:
\[
\tan(28^\circ) \approx 0.5317
\]
\[
u \approx 6 \cdot 0.5317 \approx 3.2
\]
Answer for (f): \( u \approx 3.2 \) cm
---
Final Answers:
\[
\boxed{6.2, 13.3, 9.2, 5.6, 6.7, 3.2}
\]
Parent Tip: Review the logic above to help your child master the concept of trig worksheet.
