**Objectives**

- Define the sine, cosine, and tangent functions.
- Calculate the sines, cosines, and tangents of the angles of special right triangles.
- Use trigonometric ratios to solve real-world problems.

**Key Terms**

- Cosine (COS) – In a right triangle, the ratio of the length of the angle’s adjacent leg to the length of the hypotenuse.
- Sine (SIN) – In a right triangle, the ratio of the length of the angle’s opposite leg to the length of the hypotenuse.
- Tangent (TAN) – In a right triangle, the ratio of the length of the angle’s opposite leg to the length of the adjacent leg.
- Trigonometric Ratios – The ratios of the lengths of the sides of a right triangle. The three basic trigonometric ratios are sine, cosine, and tangent.

**Review**

- The hypotenuse is the side opposite the right angle.
- The legs are the two sides opposite the acute angles.
- In Special Right Triangles, we learned the formulas 30-60-90 and 45-45-90
- See how the legs and hypotenuses relate to one another, below:

**Notes**

- In trigonometry
- θ (theta) is for unknown angles
- x is for unknown sides

- Labeling a Triangle Based on an Acute Angle
- For angle θ (theta), notice which side is labeled as “opposite” and which side is labeled as “adjacent”
- The side touching the angle is always adjacent
- The side across from the angle is always opposite
- The hypotenuse never changes. It is always across from the 90° angle.

- For angle θ (theta), notice which side is labeled as “opposite” and which side is labeled as “adjacent”

- In this animation, notice which sides are Sine, Cosine, and Tangent from angle θ (theta).

- Trigonometric Relationship Chart

- Trigonometric Functions of Special Right Triangles

- Trigonometric Functions Chart

- Example 1: What is the tangent of 67°?

Answer:

- Example 2: What is the approximate value of x in the diagram below?
- Hint: you will need to use one of the trigonometric ratios given in the box below.
- Requirement: set your calculator to “degrees”

- Solution: Note that we do NOT have a hypotenuse. We will need to use Tangent, since
- Tangent of 36° or Tangent of 54° will work.
- Tan 36° is given in the box above (let’s use that)!
- Tan 36° = 0.727

- Tan 36° also equals
- Therefore, 0.727 must equal

- Tan 36° is given in the box above (let’s use that)!
- Solve for x by multiplying x on both sides, then dividing both sides by 0.727.
- 0.727x = 23

- Answer: 31.64

- Tangent of 36° or Tangent of 54° will work.