Trigonometric Ratios and Triangles - Trigonometric Functions - High School Algebra II Unlocked (2016)

High School Algebra II Unlocked (2016)

Chapter 4. Trigonometric Functions

GOALS

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


•Use the relationship between radius, radian measure, and arc length to solve for any one of these values for a given circle and central angle

•Convert between degrees and radians for any angle measure

•Find the value of the sine, cosine, tangent, cosecant, secant, or cotangent, where it exists, for any real number

•Graph sine, cosine, and tangent functions using information about the period, midline, amplitude, horizontal shift, and any reflection, and write an equation to describe a given sine, cosine, or tangent function

•Prove and apply trigonometric function identities

•Write and use sine and cosine functions to represent real-life situations and solve problems

Lesson 4.1. Trigonometric Ratios and Triangles

REVIEW

Trigonometric functions can be expressed as ratios of side lengths in a right triangle. For an acute angle θ:

Sine: sin θ =

Cosine: cos θ =

Tangent: tan θ =

The Pythagorean theorem states that the sum of the squares of the leg lengths of a right triangle is equal to the square of its hypotenuse: a2 + b2 = c2.

Let’s review the basic trigonometric relationships as ratios of side lengths in right triangle ABC, shown below.

The hypotenuse is side AB, so sin ∠A = , cos ∠A = , and tan ∠A = . Looking at the other acute angle, sin ∠B = , cos ∠B = , and tan ∠B = .

Notice that sin ∠A = cos ∠B and cos ∠A = sin ∠B. The sine of an acute angle is always equal to the cosine of its complementary angle.

These trigonometric ratios are useful for solving problems involving right triangles, as you learned in Geometry.

Paul wants to know how tall a certain cell phone tower is. Paul walks 10 feet from the cell phone tower, lies on the ground, and measures an angle of 65° from the ground to the top of the tower. Approximately how tall is the cell phone tower from the ground to its top?

We want to find the height of the cell phone tower, which is the leg of the right triangle that is opposite the 65° angle. The given length of 10 feet is the leg adjacent to the 65° angle. The trigonometric function that relates the opposite to the adjacent side is the tangent function. Be sure to put the opposite side length in the numerator and the adjacent side length in the denominator to represent the tangent.

tan 65° = h/10

(h represents the height of the cell phone tower, in feet)

10 tan 65° = h

Multiply both sides by 10.

10(2.1445….) = h

Use your calculator to find tan 65°.

21 ≈ h

Multiply and round the product to the nearest whole number.

The cell phone tower is about 21 feet tall.

There are six basic trigonometric functions, including sine, cosine, and tangent, and their reciprocals, cosecant, secant, and cotangent.

Cosecant: csc θ =

Secant: sec θ =

Cotangent: cot θ =

In the right triangle shown below, csc θ is 13/12. What is cos θ?

The cosecant is the reciprocal of the sine of an angle, and vice versa. We know that csc θ = 13/12, so sin θ = 12/13. However, the question asks for the cosine of θ.

Let’s use the given information to label the sides of the triangle. Because csc θ = 13/12, we know that the ratio of the hypotenuse to the opposite side is 13 to 12. Label the hypotenuse as 13 and the side opposite θ as 12.

While you can work
with 12 and 13
as representative
numbers, keep in
mind that these may
not be the actual side
lengths. Trigonometric
functions are only
ratios, so it could be
that the opposite
side is 24 and the
hypotenuse is 26,
or any other pair of
real numbers with
the same ratio.

The length of the leg adjacent to θ is unknown, but we can solve for it using the Pythagorean theorem.

a2 + 122 = 132

a2 + 144 = 169

Find both squares.

a2 = 25

Subtract 144 from both sides.

a = 5

Solve for the positive value of a. Side lengths are always positive.

The missing side length is 5 units. Use this to find cos θ.

cos θ = = 5/13