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
- Height (opposite side) meters
2. Formula:
The length of the ladder (hypotenuse, ) can be found using the sine formula:
3. Substituting the known values:
4. Solve for , rearrange the formula:
Using the value :
metres
Thus, the length of the ladder is 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 = 12 cm
- Angle = 53°
2. Relationship:
- , where is the hypotenuse.
3. Rearrange for (h):
4. Substitute values:
Thus, the length of the hypotenuse is approximately 19.94 centimetres.