- Chat with library staff now
- Contact your library

Skip to Main Content
# Math: Basic Tutorials : Primary Trigonometric Ratios

## Primary Trigonometric Ratios

Right triangle trigonometry has many practical applications. For example, the ability to calculate the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or extending a tape measure along its height. In this module, we will review the primary trigonometric ratios and how they can be used to find missing side lengths or angles in a right triangle..

## Examples & Activity

### Examples

### Example 1. Find the missing side lengths. Round your final answer to the nearest tenth as needed.

### Example 2. Find the measure of angle A. Round your answer to the nearest full degree.

### Example 3. A 10 m ladder leans against a building so that the angle between the ground and the ladder is 80 degrees. How far from the building is the base of the ladder? Round your answer to the nearest tenth of a metre.

## Summary and Worksheet

## Attribution

Click on the titles below to view each example.

Image: Triangle ABC is a right triangle, where angle C is 90 degrees, and angle A is 30 degrees. Length of side a is 7, and length of sides b and c are unknown.

Solution:

Line 1: We know the angle and the opposite side, so we can use the tangent to find the adjacent side, b. Tangent theta equals opposite over adjacent.

Line 2: Substitute the given values into the equation. Tangent 30 degrees equals 7 over b.

Line 3: Solve the equation for b. b equals 7 over tangent 30 degrees.

Line 4: Simplify and round your answer to the nearest tenth. b equals 12.1.

Line 5. We can now use the sine to find the hypotenuse, c. Note you could also use the Pythagorean theorem to solve for the 3rd side. Sine theta equals opposite over hypotenuse.

Line 6. Substitute the given values into the equation. Sine 30 degrees equals 7 over c.

Line 7: Solve the equation for c. c equals 7 over sine 30 degrees.

Line 8: Simplify. c equals 14.

Line 9: Write you answer. The length of the leg is 12.1 and the length of the hypotenuse is 14.

Image: Triangle ABC is a right triangle. Angle C is 90 degrees and A is unknown and labelled as A. Length of side a is 10, and the length of side b is 4. The hypotenuse and third angle are not labelled.

Solution:

Line 1: We know the opposite and adjacent sides, so we can use the tangent to find the measure of angle A.

Line 2: Substitute the given values into the equation. Tangent of angle A equals 10 over 4.

Line 3: Solve the equation for A. Note we need to use the inverse tangent to solve for an angle. Measure of angle A equals inverse tangent of open parentheses 10 over 4 close parentheses.

Line 4: Simplify and round your answer to the nearest degree. Measure of angle A equals 68 degrees.

Line 5. Write your answer. The measure of angle A is 68 degrees.

Solution

Line 1: Draw a diagram of the situation. The diagram is a right triangle. The legs of the triangle represent the ground and the building, and the hypotenuse is the ladder and labelled with 10m. The angle between the ground and the ladder is labelled as 80 degrees. We are being asked to find the side adjacent to the 80-degree angle, so let a equal the adjacent side length.

Line 2: We know the angle and the hypotenuse, so we can use the cosine to find the adjacent side. Cosine theta equals adjacent over hypotenuse.

Line 3. Substitute the given values into the equation. Cosine 80 degrees equals a over 10.

Line 4: Solve the equation for a. a equals 10 times cosine of 80 degrees.

Line 5: Simplify and round your answer to the nearest tenth. a equals 1.7.

Line 6. Write your answer. The base of the ladder is 1.7 meters away from the wall.

- Summary: Primary Trigonometric Ratios - PDF - Opens in a new windowThis document contains a short (1 – 2 page) summary of this topic as well as detailed examples to illustrate key concepts. Use this summary to review this topic.
- Worksheet: Primary Trigonometric Worksheet - PDF - Opens in a new windowThis document contains practice questions on this topic. Use the worksheet to test your knowledge and practice the skills learned in this module. The answers to the practice questions are provided at the end.

Examples Source: "Prealgebra - opens in a new window" by Lynn Marecek & Mary Anne Anthony-Smith is licensed under CC BY 4.0 - opens in a new window / A derivative from the original work - opens in a new window

Videos from Khan Academy fall outside of the Creative Commons license of The Learning Portal.