Understanding trigonometric ratios is a fundamental part of trigonometry, a branch of mathematics that deals with the relationships between the sides and angles of triangles. This section focuses on the basic trigonometric ratios: sine (sin), cosine (cos), and tangent (tan), which are essential for solving problems related to right-angled triangles. By mastering these concepts, students can easily calculate unknown sides or angles, paving the way for more complex applications in geometry and beyond.

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Introduction to Trigonometric Ratios

In any right-angled triangle, the three primary trigonometric ratios are defined as follows:

• Sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Calculating Sides using Trigonometric Ratios

Example 1: Finding an Unknown Side

Problem: In a right-angled triangle, if the angle adjacent to the side you are trying to find is 30°, and the hypotenuse is 10 cm long, calculate the length of the adjacent side.

Solution:

1. Identify the known values: Angle = 30°, Hypotenuse = 10 cm.

2. Choose the appropriate trigonometric ratio: Since we have the hypotenuse and need to find the adjacent side, we use the cosine ratio.

3. Rearrange the formula to solve for the unknown side:

4. Calculate:

(Note: The cosine of 30° is approximately 0.866)

Example 2: Calculating the Opposite Side

Problem: Find the length of the opposite side in a right-angled triangle if the angle opposite the side is 45° and the hypotenuse is 10 cm.

Solution:

1. Identify the known values: Angle = 45°, Hypotenuse = 10 cm.

2. Select the trigonometric ratio: We use the sine ratio since we have the hypotenuse and need to find the opposite side.

3. Rearrange the formula:

4. Calculate:

(Note: The sine of 45° is approximately 0.707)

Calculating Angles using Trigonometric Ratios

To find an angle in a right-angled triangle when we know the lengths of two sides, we can use the inverse trigonometric functions: arcsin, arccos, and arctan.

Example 3: Finding an Angle

Problem: Calculate the angle opposite the side of length 7 cm in a right-angled triangle with a hypotenuse of 10 cm.

Solution:

1. Identify the known values: Opposite side = 7 cm, Hypotenuse = 10 cm.

2. Choose the trigonometric ratio: The sine ratio is suitable here.

3. Insert the known values and solve for the angle:

4. Calculate the angle using the inverse sine function:

Practice Questions

Question 1

A ladder is leaning against a wall, touching the wall at a height of 6 meters. If the angle between the ground and the ladder is 60°, find the length of the ladder.

Solution:

1. Given:

• Angle $\theta = 60^\circ$
• Height (opposite side) $h = 6$ meters

2. Formula:

The length of the ladder (hypotenuse, $L$) can be found using the sine formula:

$\sin(\theta) = \dfrac{\text{opposite}}{\text{hypotenuse}}$

3. Substituting the known values:

$\sin(60^\circ) = \dfrac{6}{L}$

4. Solve for $L$, rearrange the formula:

$L = \dfrac{6}{\sin(60^\circ)}$

Using the value $\sin(60^\circ) = \dfrac{\sqrt{3}}{2}$:

$L = \dfrac{6}{\dfrac{\sqrt{3}}{2}}$

$L = \dfrac{6 \times 2}{\sqrt{3}}$

$L = \frac{12}{\sqrt{3}}$

$L = 4\sqrt{3}$ metres

Thus, the length of the ladder is $4\sqrt{3}$ metres.

Question 2:

In a right-angled triangle, if one of the angles is 53° and the adjacent side to this angle is 12 cm, calculate the length of the hypotenuse.

Solution:

1. Given:

• Adjacent side $(a)$ = 12 cm
• Angle $(\theta)$ = 53°

2. Relationship:

• $\cos(\theta) = \dfrac{a}{h}$, where $h$ is the hypotenuse.

3. Rearrange for (h):

• $h = \dfrac{a}{\cos(\theta)}$

4. Substitute values:

• $h = \dfrac{12}{\cos(53^\circ)}$

• $h \approx 19.94 \text{cm}$

Thus, the length of the hypotenuse is approximately 19.94 centimetres.