Example 1: Identifying sides of a triangle.

In a right-angled triangle, identify the opposite, adjacent, and hypotenuse with respect to angle \( \theta \).

Solution: The hypotenuse is the longest side opposite the right angle. The side opposite angle \( \theta \) is the opposite side, and the other side next to \( \theta \) is the adjacent side.

Example 2: Using sine to find a side.

Given \( \theta = 30^\circ \) and hypotenuse = 10 cm, find the opposite side.

Solution: Use \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

\( \sin(30^\circ) = \frac{x}{10} \Rightarrow \frac{1}{2} = \frac{x}{10} \Rightarrow x = 5 \, \text{cm} \)

Example 3: Using cosine to find a side.

Given \( \theta = 60^\circ \) and hypotenuse = 12 cm, find the adjacent side.

Solution: \( \cos(60^\circ) = \frac{x}{12} \Rightarrow \frac{1}{2} = \frac{x}{12} \Rightarrow x = 6 \, \text{cm} \)

Example 4: Using tangent to find a side.

If \( \theta = 45^\circ \) and adjacent side = 8 cm, find the opposite side.

Solution: \( \tan(45^\circ) = \frac{x}{8} \Rightarrow 1 = \frac{x}{8} \Rightarrow x = 8 \, \text{cm} \)

Example 5: Finding an angle using sine.

If opposite = 4 cm and hypotenuse = 5 cm, find \( \theta \).

Solution: \( \sin(\theta) = \frac{4}{5} \Rightarrow \theta = \sin^{-1}\left(\frac{4}{5}\right) \approx 53.13^\circ \)

Example 6: Finding an angle using cosine.

If adjacent = 3 cm and hypotenuse = 5 cm, find \( \theta \).

Solution: \( \cos(\theta) = \frac{3}{5} \Rightarrow \theta = \cos^{-1}\left(\frac{3}{5}\right) \approx 53.13^\circ \)

Example 7: Finding an angle using tangent.

If opposite = 7 cm and adjacent = 24 cm, find \( \theta \).

Solution: \( \tan(\theta) = \frac{7}{24} \Rightarrow \theta = \tan^{-1}\left(\frac{7}{24}\right) \approx 16.26^\circ \)

Example 8: Real-life application of trigonometry.

A ladder leans against a wall forming an angle of \( 60^\circ \) with the ground. If the ladder is 10 m long, how high up the wall does it reach?

Solution: Use sine: \( \sin(60^\circ) = \frac{\text{opposite}}{10} \Rightarrow \frac{\sqrt{3}}{2} = \frac{x}{10} \Rightarrow x = 5\sqrt{3} \approx 8.66 \, \text{m} \)

Example 9: Solving for a missing side using Pythagoras and trigonometry.

Given: adjacent = 9 cm, angle = \( 30^\circ \). Find hypotenuse.

Solution: Use \( \cos(30^\circ) = \frac{9}{x} \Rightarrow x = \frac{9}{\cos(30^\circ)} \approx \frac{9}{0.866} \approx 10.39 \, \text{cm} \)

Example 10: Verifying trigonometric values.

Verify that for \( \theta = 45^\circ \), \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Solution: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \Rightarrow \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 = \tan(45^\circ) \)