# 4.1 – Trigonometric Ratios

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.

• 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: $\frac{12}{5}$

• 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 $Tan=\frac{Opp}{Adj}$
• 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 $\frac{23}{x}$
• Therefore, 0.727 must equal $\frac{23}{x}$
• Solve for x by multiplying x on both sides, then dividing both sides by 0.727.
• 0.727x = 23
• $x=\frac{23}{0.727}$