This section uses the trigonometric ratios to find sides and the hypotenuse of right triangles. A calculator is needed.

The three trigonometric ratios — sine, cosine, and tangent — can be used to find missing side lengths or missing angle measures.

Example

Fill in the blanks.

Step 1: Identify the reference angle, the adjacent leg, the opposite leg, and the hypotenuse.

Mouse over to reveal each term:

Step 2: Identify the information you are given and the information you are looking for.

• Given: ___________
Interactive popup. Assistance may be required.
Check Your Answer
Given: The length of the hypotenuse

• Find: ____________
Interactive popup. Assistance may be required.
Check Your Answer
Find: The length of the adjacent leg

Step 3: Identify the trigonometric ratio that relates the two side lengths from Step 2.

Interactive popup. Assistance may be required.

Check Your Answer

The cosine ratio relates the length of the hypotenuse and the length of the adjacent leg.

Step 4: Set up a ratio, including the angle measure, known side length, and unknown side length.

Interactive popup. Assistance may be required.

Check Your Answer

cosL =   Adjacent Leg Hypotenuse cos(65°) =   x 14

Step 5: Solve for x. Remember that cos (65°) is a number, just like 4, 19, or 0.45. Round to the appropriate place value.

Interactive popup. Assistance may be required.

Check Your Answer

Example 2

Step 1: Identify the reference angle, the adjacent leg, the opposite leg, and the hypotenuse.

Mouse over to reveal each term.

Step 2: Identify the information you are given and the information you are looking for.

• Given: ___________
Interactive popup. Assistance may be required.
Check Your Answer
Given: The length of the opposite leg
• Find: ____________
Interactive popup. Assistance may be required.
Check Your Answer
Find: The length of the hypotenuse

Step 3: Identify the trigonometric ratio that relates the two side lengths from Step 2.

Interactive popup. Assistance may be required.

Check Your Answer

The sine ratio relates the length of the opposite leg and the length of the hypotenuse.

Step 4: Set up a ratio, including the angle measure, known side length, and unknown side length

Interactive popup. Assistance may be required.

Check Your Answer

sinQ =   Opposite Leg Hypotenuse sin(35°) =   20 x

Step 5: Solve for x. Remember that sin (35°) is a number, just like 4, 19, or 0.45. Round to the appropriate place value.

Interactive popup. Assistance may be required.

Check Your Answer