VannKit Geometry - Right Triangle: Trig Ratio- SOH.CAH.TOA

Trigonometry is an important tool for evaluating measurements of height and distance. It plays an important role in surveying, navigation, engineering, astronomy and many other branches of physical science.

Basic Trigonometry involves the ratios of the sides of right triangles. The three ratios are called tangent, sine and cosine. It can then be extended to other ratios and Trigonometry in the Cartesian Plane.

Identifying Hypotenuse, Opposite and Adjacent Side

Note: The adjacent and the opposite sides depend on the angle θ. For complementary angle of θ, the labels of the 2 sides are reversed.

Sides Of A Right Triangle

Hypotenuse, Adjacent and Opposite Sides.

In the right triangle PQR,

the side PQ, which is opposite to the right angle PRQ is called the hypotenuse. The hypotenuse is the longest side of the right triangle.
the side RQ is called the adjacent side of angle θ.
the side PR is called the opposite side of angle θ.

Example 1: Hypotenuse, Opposite & Adjacent

Identify the hypotenuse, adjacent side, and opposite side in the following triangle:a) for angle xb) for angle y

Solution:

a) For angle x: AB is the hypotenuse, AC is the adjacent side , and BC is the opposite side.

b) For angle y: AB is the hypotenuse, BC is the adjacent side , and AC is the opposite side.

Trigonometric Functions / Trigonometric Ratios

Now that we are able to name the sides of right triangles, the basics of trigonometric functions (or trigonometric ratios) can now be introduced. Trigonometric ratios establish a relationship (or a link) between the angles of a right triangle with its sides. The first six trig functions are...

- Sine,
- Cosine and
- Tangent

Trigonometric Functions: Sine of an Angle

The sine θ is equal to the opposite side divided by the hypotenuse side.

Example 2: Sine

Calculate the value of sin Z in the triangle shown.

Solution: Opposite side = 21, Hypotenuse = 35

Sin Z = opposite/hypotenuse = 21/35

Trigonometric Functions: Cosine of an Angle

The cosine θ is equal to adjacent side divided by the hypotenuse side.

Example 3: Cosine

Calculate the value of cos Z in the triangle shown.

Solution: adjacent side = 28, Hypotenuse = 35

Sin Z = opposite/hypotenuse = 28/35

Trigonometric Functions: Tangent

The tangent θ is equal to the opposite side divided by the hypotenuse side.

Example 4: Tangent

Calculate the value of Tan Z in the triangle shown.

Solution: opposite side = 21, adjacent = 28

Tan Z = opposite/adjacent = 21/28

Activity 1: Poll

Activity 2: Trigonometric Ratios

Introducing the First Six Trig Ratio: SOHCAHTOA

"SOHCAHTOA" is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.

Solving problems using SOHCAHTOA

"SOHCAHTOA" is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.

Activity 3: Trigonometry Vocabulary 1

Activity 4: Trigonometry Vocabulary 2 (Game Show)

Expressing Trig Ratios as Decimals

Trig ratio can easily be expressed as decimal by simply using a calculator to do the conversion.