How to Enter Cos 90 Degrees on Calculator

Calculating the cosine of 90 degrees is a fundamental trigonometric operation that appears in many mathematical and scientific applications. This guide explains how to perform this calculation using both calculator methods and manual techniques, along with important considerations and common pitfalls.

How to Calculate cos 90 Degrees

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For 90 degrees, this relationship becomes particularly simple because of the properties of right triangles.

Formula

cos(θ) = adjacent/hypotenuse

For θ = 90°:

cos(90°) = adjacent/hypotenuse = 0

This is because in a right triangle, when the angle is 90 degrees, the adjacent side to that angle has zero length (it's the side opposite the right angle). Therefore, the cosine of 90 degrees is always 0.

Step-by-Step Guide

Using a Calculator

Turn on your calculator and ensure it's in degree mode (not radian mode).
Enter the number 90.
Press the cosine (cos) function button.
Press the equals (=) button to get the result.

Manual Calculation

Draw a right triangle with one angle measuring 90 degrees.
Identify the side adjacent to the 90-degree angle (which will be 0 length).
Measure the hypotenuse (any side of the triangle that is not the adjacent side).
Divide the length of the adjacent side (0) by the hypotenuse to get cos(90°) = 0.

Using a Calculator

Most scientific calculators have a cosine function that can compute cos(90°) directly. Here's how to do it:

Note: Ensure your calculator is in degree mode before entering the calculation. If it's in radian mode, you'll get a different result.

1. Press the "DEG" button to ensure the calculator is in degree mode.

2. Enter the number 90.

3. Press the "cos" button.

4. Press the "=" button to display the result.

The calculator should display 0 as the result.

Manual Calculation

To calculate cos(90°) manually, you can use the definition of cosine in a right triangle:

cos(θ) = adjacent/hypotenuse

For a right triangle with a 90° angle:

The side adjacent to the 90° angle is the side opposite the right angle, which has length 0.
The hypotenuse is any side of the triangle that is not the adjacent side.

Therefore, cos(90°) = 0 / hypotenuse = 0.

Common Mistakes

When calculating cos(90°), there are several common mistakes to avoid:

Using radian mode: If your calculator is in radian mode, it will calculate cos(90 radians) instead of cos(90 degrees), resulting in a different (and incorrect) value.
Incorrect triangle construction: When drawing the triangle manually, ensure you're correctly identifying the adjacent side as the one opposite the right angle.
Forgetting the definition: Remember that cosine is defined as adjacent/hypotenuse, not any other ratio in the triangle.

FAQ

Why is cos(90°) equal to 0?: Because in a right triangle, the side adjacent to a 90° angle has zero length (it's the side opposite the right angle).
What happens if I calculate cos(90°) in radian mode?: You'll get a different result (approximately -0.98999) because 90 radians is a much larger angle than 90 degrees.
Can I use a calculator to find cos(90°)?: Yes, but make sure your calculator is in degree mode. Most scientific calculators have a degree mode setting.
Is cos(90°) the same in all right triangles?: Yes, because in any right triangle with a 90° angle, the adjacent side is always 0, making cos(90°) = 0.
What's the difference between cos(90°) and sin(90°)?: cos(90°) = 0, while sin(90°) = 1 because sin(θ) = opposite/hypotenuse, and the opposite side to 90° is the hypotenuse.