Trigonometric Ratio Calculator

Published on: December 29, 2024
Final Answer: Free • Full Steps: Plus

This Trigonometric Ratio Calculator works out the sine, cosine or tangent of an angle from two side lengths of a right triangle, and shows the working line by line. Each ratio pairs the angle with two named sides, so the first decision is which pair the chosen function needs. Once the two lengths are in place the ratio is a single division, and the answer is a plain number rather than a length. Following the same pattern every time keeps the three ratios from blurring together.

Step-by-step method

  1. Set up the problem.
  2. Write the ratio for the chosen function.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.

Formulas:

\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
See Example 1 Hide Example 1

Problem

\(\cos(\theta),\; \text{adjacent} = 8,\; \text{hypotenuse} = 17\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the ratio for the chosen function.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the cosine ratio needs are \(\text{adjacent} = 8\) and \(\text{hypotenuse} = 17\).

\(\text{adjacent} = 8,\; \text{hypotenuse} = 17\)

Step 2

Step 2 - Write the ratio for the chosen function.

In this problem: The chosen function is cosine, which compares the adjacent with the hypotenuse, giving \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{adjacent}\) with \(8\) and \(\text{hypotenuse}\) with \(17\) gives \(\cos(\theta) = \frac{8}{17}\).

\(\cos(\theta) = \frac{8}{17}\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(8\) by \(17\) gives \(0.47\).

\(\cos(\theta) = \frac{8}{17} \approx 0.47\)

Final Answer

\(\cos(\theta) \approx 0.47\)
See Example 2 Hide Example 2

Problem

\(\tan(\theta),\; \text{opposite} = 5,\; \text{adjacent} = 12\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the ratio for the chosen function.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the tangent ratio needs are \(\text{opposite} = 5\) and \(\text{adjacent} = 12\).

\(\text{opposite} = 5,\; \text{adjacent} = 12\)

Step 2

Step 2 - Write the ratio for the chosen function.

In this problem: The chosen function is tangent, which compares the opposite with the adjacent, giving \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{opposite}\) with \(5\) and \(\text{adjacent}\) with \(12\) gives \(\tan(\theta) = \frac{5}{12}\).

\(\tan(\theta) = \frac{5}{12}\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(5\) by \(12\) gives \(0.42\).

\(\tan(\theta) = \frac{5}{12} \approx 0.42\)

Final Answer

\(\tan(\theta) \approx 0.42\)

Example Problems

Click one to load it into the box and solve it.