Right Triangle Trigonometry

Pythagorean Theorem

Examples

Example 1

Find the missing side of a right triangle with legs 8 units and 13 units.

Example 2

Find the hypotenuse of a right triangle with legs measuring 27 units and 22 units.

Example 3

The hypotenuse of a right triangle is 9 units and one leg of the triangle is 7 units. What is the measurement of the other leg?

Example 4

The hypotenuse of a right triangle is 30.5 units and one leg is 19.1 units. What is the length of the other leg?

Example 5

An isosceles triangle has a base of 22 units and a side of 24 units. What is the height of the isosceles triangle?

Example 6

A trapezoid has bases of 18 units and 5 units. One side is 16 units. The other side is unknown, but is perpendicular to both bases. What is the length of that side?

Example 7

A triangle has sides 14 units, 29 units, and a base of x units. The height of the triangle is 12 units. What is the length of the base?


Special Triangles

45°-45°-90° Special Right Triangle

The length of the hypotenuse of a right triangle with two angles of 45 degrees is √2 times as long as the sides opposite the 45° angles.

Examples

Example 1

A right triangle has another angle that is 45°. The side opposite the 45 degree angle is 8 units. The side opposite the unknown angle is x units. The hypotenuse is y units in length. Find the value of x and y.

Example 2

A right triangle has another angle that is 45°. The side opposite the 45 degree angle is 25 units. The side opposite the unknown angle is y units. The hypotenuse is x units in length. Find the value of x and y.

Example 3

A right triangle has another angle that is 45°. The hypotenuse is 19 units long. The other sides are labeled x and y. Find the values of x and y.

30°-60°-90° Special Right Triangle

A 30°-60°-90° special right triangle has a hypotenuse twice as long as the shortest side. The shortest side is opposite the 30° angle. The side opposite the 60° angle is √3 times as long as the shortest side.

Examples

Example 1

A right triangle has another angle that measures 60°. The shortest side of the triangle is 5 units long. Find the lengths of the other sides.

Example 2

A right triangle has another angle that measures 30°. The side opposite the 30° angle is 14 units long. Find the lengths of the other sides.

Example 3

A right triangle has another angle that measures 30°. The hypotenuse of the triangle is 32 units long. Find the lengths of the other sides.


Trig Ratios

In this section, we’ll study the three main trig ratios, sine, cosine, and tangent. They are defined as follows:

Many people remember this with the acronym SOHCAHTOA.

Examples

Example 1

Find the sine, cosine, and tangent of the angles that are not the right angle in a right triangle with side lengths 5, 12, and 13.

Example 2

Find the sine, cosine, and tangent of the angles that are not the right angle in a right triangle with side lengths 9, 12, and 15.

Example 3

Find the sine, cosine, and tangent of the angles that are not the right angle in a right triangle with side lengths 16, 30, and 34.


Trig Ratios to Find Missing Sides

We can also use the trig ratios to find the length of missing sides in a right triangle.

Examples

Example 1

A right triangle has another angle measuring 48°. The side next to 48° measures 17 units. Find the lengths of the other sides in the triangle.

Example 2

A right triangle has a hypotenuse of length 29 units and an angle measuring 67°. Find the lengths of the other sides.

Example 3

A right triangle has another angle measuring 29°. The side opposite 29° measures 12 units. Find the lengths of the other sides.


Trig Ratios to Find Missing Angles

You can also use the inverse of sine, cosine, and tangent (the notation for these is sin−1, cos−1, and tan−1). You may also see these referred to as arcsin, arccos, and arctan.

Examples

Example 1

A right triangle has a hypotenuse measuring 24 units and a leg measuring 15 units. Find all the angle measurements in the triangle.

Example 2

A right triangle has a hypotenuse measuring 11 units and a leg measuring 8 units. Find all the angle measurements in the triangle.

Example 3

A right triangle has two legs measuring 20 units and 37 units. Find all the angle measurements in the triangle.

Note: These notes were prepped by a Tim.