**Introduction to Trigonometry Basics**

**Lesson Summary**

In this lesson, you will learn how to write the trigonometric ratios sine, cosine, and tangent for a right triangle. You will learn how to use the trigonometry ratios to find unknown angle or side measurements.

**P**roperties of similar right triangles are the basis for trigonometry. Measurements sometimes cannot be made by using a measuring tape. For example, the distance from a ship to an airplane can't be determined this way; however, it could be found by using trigonometric ratios. The word *trigonometry* comes from the Greek language; it means "triangle measurement."

Recall that the hypotenuse is the side of the triangle across from the right angle. The other two sides of the triangle are called legs. These two legs have special names in relation to a given angle: adjacent leg and opposite leg. The word *adjacent* means *beside*, so the adjacent leg is the leg beside the angle. The opposite leg is across from the angle. You can see in the following figure how the legs are named depending on which acute angle is selected.

Sine, cosine, and tangent ratios for the acute angle A in a right triangle are as follows:

sin *A* =

cos *A* =

tan *A* =

**Examples:**

Express each ratio as a decimal to the nearest thousandth.

- sin
*A*, cos*A*, tan*A* - sin
*B*, cos*B*, tan*B*

**Solution:**

**Using a Trigonometric Table**

Trigonometric ratios for all acute angles are commonly listed in tables. Scientific calculators also have functions for the trigonometric ratios. Consult your calculator handbook to make sure you have your calculator in the *degree* and not the *radian* setting. Part of a trigonometric table is given here.

**Example:**

Find each value.

- cos 44°
- tan 42°

**Solution:**

- cos 44° = 0.719
- tan 42° = 0.900

**Example:**

Find *m**A*.

- sin A = 0.656
- cos A = 0.731

**Solution:**

*m**A*= 41°*m**A*= 43°

**Finding the Measure of an Acute Angle**

The trigonometric ratio used to find the measure of an acute angle of a right triangle depends on which side lengths are known.

**Example: **

Find *m**A*.

**Solution: **

The sin A involves the two lengths known.

Note that the ≈ symbol is used in the previous solution because the decimal 0.313 has been rounded to the nearest thousandths place.

Using the sin column of a trigonometric table, you'll find:

sin 18° = 0.309 } difference 0.004

sin A ≈ 0.313

sin 19° = 0.326 } difference 0.013

Since 0.313 is closer to 0.309, *m**A* ≈ 18° to the nearest degree.

A scientific calculator typically includes a button that reads sin^{–1}, which means *inverse sine*, and can be used to find an angle measure when the sine is known.

**Example: **

Find *m**C*.

**Solution: **

The tan *C* involves the two lengths known.

- Using the tan column, you'll find:

- tan 36 = 0.727

- Therefore,

*m*

*C*= 36°.

Again, the tan^{–1} button on a scientific calculator can be used to find an angle measure when its tangent is known.

**Finding the Measure of a Side**

The trigonometric ratio used to find the length of a side of a right triangle depends on which side length and angle are known.

**Example: **

Find the value of *x* to the nearest tenth.

Practice problems for these concepts can be found at: Trigonometry Basics Practice Questions.

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