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

High School Algebra II Unlocked (2016)

Chapter 4. Trigonometric Functions

Lesson 4.3. Trigonometric Functions and Circles

REVIEW

A 45-45-90 triangle is a right triangle with angle measures of 45°, 45°, and 90°, and which has side lengths in the ratio 1:1: , respectively opposite each of those angles.

When we looked at trigonometric functions in relationship to triangles, we could only define them for angles between 0 and 90 degrees, or 0 and π/2 radians.

Using a unit circle in the coordinate plane allows us to evaluate trigonometric functions (where they exist) for all real numbers. We will express these real numbers in terms of radian measures of central angles of the unit circle, measured counterclockwise from the positive x-axis.

This means that one side of the angle is always fixed on the positive x-axis. This is the initial side of the angle. The terminal side is the other side forming the angle, and it can be rotated anywhere along the circle. Both the initial side and the terminal side have a length of 1, because they are both radii of the unit circle. Let’s look at angles with measures of π/4, π/2, π, 5π/4, and 9π/4.

The initial side is always stationary. The terminal side rotates counterclockwise the given number of radians to form the angle. An angle of measure π/2 is a right angle, and an angle of measure π is a straight angle. An angle of measure 2π (not shown) is a full revolution of the circle, so any angle with a greater measure continues into a second revolution of the circle, as with 9π/4. This angle looks like π/4 because it is coterminal with π/4, meaning that their terminal sides are the same. This is because 9π/4 = 2π + π/4: a full revolution plus an additional rotation of π/4.

Angles defined this way can also have negative measures, which represent a clockwise rotation of the terminal side.

Notice that a −3π/4 angle is coterminal with a 5π/4 angle, and a −π angle is coterminal with a π angle.

Coterminal angles have a difference of measure that is a multiple of 2π.

This holds true for each of the three coterminal angle pairs that we have looked at so far.

π − (−π) = π + π = 2π

5π/4 − (−3π/4) = 5π/4 + 3π/4 = 8π/4 = 2π

9π/4 − π/4 = 8π/4 = 2π

If we looked at the angle 5π, it would be coterminal with π. The difference between 5π and π is 4π, which is 2 times 2π. An angle of 5π makes two full revolutions of the unit circle and then rotates an additional π radians to form a straight angle.

For any angle measure, you can determine the trigonometric function value, if it exists, using the unit circle. The sine and cosine exist for all real numbers.

Using central angles formed by radii of a unit circle, find the values of sin π/4, sin π/2, sin 3π/4, sin π, and sin 7π/6.

Let’s look at the angle π/4, adding a vertical line to complete a right triangle with the terminal side and part of the initial side.

The sine value is . Any radius of a unit circle is 1 unit long, so the hypotenuse of this right triangle is 1, which means the sine is equal to the opposite (vertical) side. So, the sine of π/4 is the y-value of the point where the terminal side of the angle reaches the unit circle. The degree equivalent of π/4 is 45°, so this is a 45-45-90 triangle, which means that the leg-to-hypotenuse ratio is 1: . Let’s set up a proportion with leg lengths in the numerators and hypotenuse lengths in the denominators.

y/1 = 1/

The ratio of opposite leg length to radius 1 is equal to the ratio 1: .

y = 1/

Rewrite y/1 as y.

y = /2

Multiply by / to rationalize the denominator.

The height of the endpoint of the terminal side of the angle is /2, so the sine of π/4 is /2.

For any central angle, you can draw a vertical line to form a right triangle. The sine of the angle is equal to the y-value of the point where the terminal side meets the unit circle, because the hypotenuse of the right triangle formed is always 1.

The simplicity
of trigonometric
functions in relation
to the unit circle is
another reason why
mathematicians like
the unit circle so much.

So, sin θ ranges from 0 when θ = 0 to 1 when θ = π/2, as indicated by the right triangles of increasing height shown in the following figure.

For a central angle of π/2, the leg “opposite” the central angle in the right triangle is actually the same length as the hypotenuse, 1. This is not literally a triangle at this point, because the adjacent side (the horizontal leg) has shrunk to 0, but the sine value still exists for π/2. So, sin π/2 = 1.

Here is the central angle 3π/4. Notice that we still draw a vertical line from the x-axis to the end of the terminal side of the angle to form a right triangle. Now the sine is calculated using the acute angle of the triangle, even though we are finding the sine of 3π/4.

The sine of 3π/4 is the sine value of the acute angle that is supplementary to 3π/4, or π/4. Again, the sine is equal to the y-value of the point where the terminal side of the angle meets the unit circle, because this is the opposite side of the right triangle formed with the x-axis and terminal side. So, sin 3π/4 = /2.

The sine values are decreasing in this quadrant, for angle measures from π/2 to π. The central angle π is a straight angle, or a straight line, as shown below.

For a straight angle, the height of the opposite leg is 0, and the y-value of the endpoint of the terminal side is 0. So, sin π = 0.

The angle 7π/6 is shown below.

Here, the right triangle is below the x-axis, where y-values are negative. The sine of the central angle is still equal to the y-value of the point where the terminal side meets the unit circle, so in this case it is negative.

The angle 7π/6 is a straight angle (π) plus an additional π/6 angle. The degree equivalent of π/6 is 30°, so the right triangle shown is a 30-60-90 triangle. The side opposite the π/6 angle has a length that is 1/2 the hypotenuse. The hypotenuse is 1 unit long, so the opposite leg is 1/2 unit long, and the y-value of the endpoint is −1/2. So, sin 7π/6 = −1/2.

For all angles greater than π and less than 2π, the sine value is negative. From π to 3π/2, the sine values decrease from 0 to −1, and from 3π/2 to 2π, the sine values increase from −1 to 0.

Remember that coterminal angles have the same terminal side, so they also have the same sine value. For example, sin (−5π/6) = sin 7π/6 = −1/2.

In fact, because coterminal angles share the same terminal side, they have the same function value for each of the trigonometric functions.