Trigonometry Worksheets | Practice Questions and Answers | Cazoomy - Free Printable

To solve for the missing angles in the given right triangles, we will use trigonometric ratios. The three primary trigonometric ratios are:

- Sine (sin): $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
- Cosine (cos): $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
- Tangent (tan): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

We will determine which ratio to use based on the sides given in each triangle.

---

Triangle (a)

- Given: Opposite side = 3 cm, Adjacent side = 6 cm
- Unknown angle: $x$

Since we have the opposite and adjacent sides, we use the tangent ratio:
\[
\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{6} = \frac{1}{2}
\]
To find $x$, we take the inverse tangent (arctangent):
\[
x = \tan^{-1}\left(\frac{1}{2}\right)
\]
Using a calculator:
\[
x \approx 26.57^\circ
\]

---

Triangle (b)

- Given: Opposite side = 4 cm, Hypotenuse = 15 cm
- Unknown angle: $x$

Since we have the opposite side and the hypotenuse, we use the sine ratio:
\[
\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{15}
\]
To find $x$, we take the inverse sine (arcsine):
\[
x = \sin^{-1}\left(\frac{4}{15}\right)
\]
Using a calculator:
\[
x \approx 15.34^\circ
\]

---

Triangle (c)

- Given: Adjacent side = 8 cm, Hypotenuse = 10 cm
- Unknown angle: $x$

Since we have the adjacent side and the hypotenuse, we use the cosine ratio:
\[
\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{10} = \frac{4}{5}
\]
To find $x$, we take the inverse cosine (arccosine):
\[
x = \cos^{-1}\left(\frac{4}{5}\right)
\]
Using a calculator:
\[
x \approx 36.87^\circ
\]

---

Triangle (d)

- Given: Opposite side = 9 cm, Adjacent side = 12 cm
- Unknown angle: $x$

Since we have the opposite and adjacent sides, we use the tangent ratio:
\[
\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{9}{12} = \frac{3}{4}
\]
To find $x$, we take the inverse tangent (arctangent):
\[
x = \tan^{-1}\left(\frac{3}{4}\right)
\]
Using a calculator:
\[
x \approx 36.87^\circ
\]

---

Triangle (e)

- Given: Opposite side = 7 cm, Adjacent side = 20 cm
- Unknown angle: $x$

Since we have the opposite and adjacent sides, we use the tangent ratio:
\[
\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{20}
\]
To find $x$, we take the inverse tangent (arctangent):
\[
x = \tan^{-1}\left(\frac{7}{20}\right)
\]
Using a calculator:
\[
x \approx 19.29^\circ
\]

---

Triangle (f)

- Given: Opposite side = 42 cm, Adjacent side = 48 cm
- Unknown angle: $x$

Since we have the opposite and adjacent sides, we use the tangent ratio:
\[
\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{42}{48} = \frac{7}{8}
\]
To find $x$, we take the inverse tangent (arctangent):
\[
x = \tan^{-1}\left(\frac{7}{8}\right)
\]
Using a calculator:
\[
x \approx 41.19^\circ
\]

---

Final Answers

\[
\boxed{
\begin{aligned}
(a) & \quad x \approx 26.57^\circ \\
(b) & \quad x \approx 15.34^\circ \\
(c) & \quad x \approx 36.87^\circ \\
(d) & \quad x \approx 36.87^\circ \\
(e) & \quad x \approx 19.29^\circ \\
(f) & \quad x \approx 41.19^\circ \\
\end{aligned}
}
\]