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

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.

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

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.

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.

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.