- Identify and use the tangent ratio in a right triangle.
- Understand tangent ratios in special right triangles.
The Tangent Ratio
The first ratio to examine when studying right triangles is the tangent.
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. The hypotenuse is not involved in the tangent at all.
Recall that a ratio is the same as a fraction, so the tangent is the fraction of the opposite side over the adjacent side.
This means that tangent is: the ____________________ side divided by the ____________________ side.
Be sure when you find a tangent that you find the opposite and adjacent sides relative to the angle in question.
You must be careful that the opposite side is across from the angle you are taking the tangent of, and the adjacent side is next to that same angle!
Like always, be sure to reduce the fraction in your final answer!
1. Fill in the blank: Another word for ratio is _______________________.
2. Which side of a right triangle is NOT used in the tangent ratio?
To find these ratios, first identify the sides opposite and adjacent to each angle:
Notice that the tangent is different for different angles because of which sides are opposite and which sides are adjacent to each angle in the triangle.
On the blank triangle above,
1. Label the right angle.
3. Label the hypotenuse.
(Note: some sides will have more than one label!)
Tangents of Special Right Triangles
It may help you to learn some of the most common values for tangent ratios. The list below shows you values for angles in special right triangles.
This means that a right isosceles triangle has one angle that is _______ and the other two that equal _______.
Likewise, both legs of the isosceles triangle are the same length! When you take the tangent of the opposite leg over the adjacent leg, the value is 1.
You can use these ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals one of these values, you can identify the measurement of the angle.
Reading Check (Challenge):
1. Which side length is the hypotenuse?