Trigonometric Ratios - Trigonometry - High School Geometry Unlocked (2016)

High School Geometry Unlocked (2016)

Chapter 4. Trigonometry

GOALS

By the end of this chapter, you will be able to:


•Understand and apply the basic trigonometric functions (sine, cosine, and tangent) and their reciprocals (secant, cosecant, and cotangent)

•Use trigonometric functions and their inverses on your calculator

•Find sine, cosine, and tangent values for complementary angles

•Use trigonometric functions to solve problems with right triangles

•Use the Pythagorean theorem to derive and understand additional trigonometric identities

•Use the Law of Sines and Law of Cosines to solve problems with non-right triangles

Lesson 4.1. Trigonometric Ratios

WHAT IS TRIGONOMETRY?

Trigonometry is the study of triangles. It’s actually so important that there is an entire branch of mathematics devoted to this topic. The most common ways that you’ll use trigonometry in high school include solving for unknown side lengths and/or angles in triangles. In the real world, trigonometry has many applications in other fields such as physics, engineering, astronomy, and even music.

Here is how you may see trigonometric ratios on the SAT.

In rectangle PQRS, shown below, the diagonal PR is 15 meters. If the sine of ∠SPR is , what is the value of RS ?

A)0.01

B)0.70

C)7.0

D)10.5

TRIGONOMETRIC RATIOS

Almost everything we know about trigonometry can be derived from relationships found in right triangles. In fact, some basics that you already know—like the Pythagorean theorem—play a fundamental role. Trigonometry allows us to use the known ratios of a triangle to solve for unknown information, like side lengths.

Looking at the triangle above, you know that one angle is 90° and one angle is 30°. What else do you know? Since the angles in a triangle always add up to 180°, you’d also be able to find x. 180° − (90° + 30°) = 60°, so x would be 60°. You have all three angles for this triangle, then, which also means you know its proportions. That’s the fundamental theorem of trigonometry—if you know the angles of a triangle, you know the ratios of the sides. That is, you might not know the actual values of the individual sides, but you know the relationships between them.

Based on what we know of triangle proportions, trigonometry has three basic functions to express these relationships.

Here, we’ll define some terms. Then, you’ll see some examples.

θ (called “theta”)—a commonly used variable for angles. (Think of it like an x.) Opposite and adjacent—the two legs of the triangle.

The opposite side—the leg that’s opposite from (i.e. not touching) the angle θ.

The adjacent side—the leg that’s adjacent to (i.e. touching) the angle θ.

The hypotenuse—the longest side of a right triangle. It’s always opposite from the 90° angle.

Sine

The sine of an angle is the ratio of its opposite side to the hypotenuse.

The sine function of an angle θ is abbreviated as sin θ.

Cosine

The cosine of an angle is the ratio of its adjacent side to the hypotenuse.

The cosine function of an angle θ is abbreviated as cos θ.

Tangent

The tangent of an angle is the ratio of its opposite side to its adjacent side.

The tangent function of an angle θ is abbreviated as tan θ.

If the sine of ∠SPR is , that would mean the diagonal of the rectangle is 10. We know that it’s 15, not 10, so we need to set up a proportion. Your proportion should look like this: = . When you solve for x, you get 10.5 meters.

The correct answer is (D).

A mnemonic for these functions is SOHCAHTOA (pronounce it like “so-ca-toe-a”).

SOHCAHTOA

Let’s take another look at the 30°-60°-90° triangle. Remember that this is a “special” right triangle, whose proportions you may have memorized previously.

For example, if the short side is 2, then the triangle would have the following side lengths.

Here are the trigonometric ratios for these angles.

Remember, “opposite”
and “adjacent” are
relative to the angle
you’re working with.
The “adjacent” of
30° is the same as
the “opposite” of
60°, and vice versa.

Fractions in simplified
form do not have
irrational numbers in
the denominator. To
simplify, multiply the
irrational number on
the top and bottom
of the fraction.

Example:

=

=

This triangle is larger than the one in the previous example. However, it is still a 30°-60°-90° triangle. In other words, the two triangles are similar—they have the same angles, even though they are different sizes.

Using these side lengths, we can see that the trigonometric ratios are still the same.

sin30° =

cos30° =

tan30° =

And so on. Therefore, since similar triangles have the same angles, they also have the same trigonometric relationships.

Each angle has given values for sine, cosine, and tangent.

Recall the AAA
Similarily Postulate
from Lesson 3.2.

Here’s our favorite right triangle, the 3-4-5. Can you determine the sin, cos, and tan values for x° and y°?

Check your answers at the end of the chapter.