Lesson The Amazing Unit Circle: Trigonometric Identities

The unit circle definition of the trigonometric functions provides a lot of information

The trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x² + y² = 1.

Start by constructing the ray from the origin at angle θ (measured counter-clockwise from the positive x-axis). This ray meets the unit circle at a point P = (x,y).

Cosine of the angle θ is defined to be the horizontal coordinate x of this point P: cos(θ) = x.

Sine of the angle θ is defined to be the vertical coordinate y of P: sin(θ) = y.

As θ changes so does the position of the point P and thus the values of cos(θ) = x and sin(θ) = y also change. In this way the functions f(&theta) = cos(θ) and g(&theta) = sin(θ) are defined.

Right away the unit circle gives us properties of the cosine and sine functions. Since the point P lies on the unit circle, both the cosine and sine functions have range -1 to 1. We also see that the points at the right, top, left and bottom of the circle give the values:

cos(0) = cos(0°) = 1 and sin(0) = sin(0°) = 0; cos(π/2) = cos(90°) = 0 and sin(π/2) = sin(90°) = 1;
cos(π) = cos(180°) = -1 and sin(π) = sin(180°) = 0; cos(3π/2) = cos(270°) = 0 and sin(3π/2) = sin(270°) = -1.

In this lesson we focus on trigonometric identities that can be immediately "read off" the unit circle.

1. The Fundamental Trigonometric Identity

Because the point P = (x,y) lies on the unit circle, these coordinates satisfy the equation x² + y² = 1. We have x = cos(θ) and y = sin(θ), and therefore

cos²(θ) + sin²(θ) = 1.

This is called the fundamental trigonometric identity.

2. Complementary Angle Identities

Two angles are complementary if they add to make a right angle (π/2 = 90°). The angle complementary to θ is the angle π/2 - θ = 90° - &theta.
To put the angle complementary to θ in standard position, start by reflecting the angle θ in the line y = x, which bisects the first quadrant. The angle BOQ is θ, so the angle AOQ measures π/2 - θ = 90° - θ. Thus Q has coordinates (cos(π/2-θ),sin((π/2-θ)) = (cos(90°-θ),sin((90°-θ)).
When a point is reflected in the line y = x, its coordinates are reversed. So Q also has coordinates (sin(θ),cos(θ)). Therefore:

cos(π/2-θ) = sin θ & sin(π/2-θ) = cos θ

or

cos(90°-θ) = sin θ & sin(90°-θ) = cos θ.

These are the complementary angle identities.

Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the complementary angle identities hold for all angles θ.

3. Supplementary Angle Identities

Two angles are supplementary if they add to make a straight angle (π = 180°). The angle supplementary to θ is the angle π - θ = 180° - &theta.

To put the angle supplementary to θ in standard position, start by reflecting the angle θ in the y-axis. The angle BOQ is θ, so the angle AOQ measures π - θ = 180° - θ. Thus Q has coordinates (cos(π-θ),sin(π-θ)) = (cos(180°-θ),sin(180°-θ)).

When a point (a,b) is reflected in the y-axis, it moves to the point (-a,b). So Q also has coordinates (-cos(θ),sin(θ)). Therefore:

cos(π-θ) = - cos θ & sin(π-θ) = sin θ

or

cos(180°-θ) = - cos θ & sin(180°-θ) = sin θ.

These are the supplementary angle identities.

Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the supplementary angle identities hold for all angles θ.

4. Phase Shift Identities

The identities in this section are about angles that differ by a right angle.

Start with the angle θ in standard position. Add a right angle to θ, so angle AOC measures θ + π/2 = θ + 90° and C has coordinates (cos(θ + π/2),sin(θ + π/2)) = (cos(θ + 90°),sin(θ + 90°)). Since angle AOB is a right angle, the angle BOC is &theta.

Rotating the angle AOP by a right angle counterclockwise around the origin shows that the length OE = OD = cos(θ) and the length EC = DP = sin(θ). Thus C also has coordinates (-sin(θ),cos(&theta)). Therefore:

cos(θ+π/2) = - sin θ & sin(θ+π/2) = cos θ

or

cos(θ+90°) = - sin θ & sin(θ+90°) = cos θ.

Two more identities can now be obtained easily.

The angle AOF is obtained by subtracting a right angle from &theta, so F has coordinates (cos(&theta - π/2),cos(&theta - π/2)). Notice that &theta - π/2 = &theta - 90° and &theta + π/2 = &theta + 90° differ by a straight angle. This means the points C and F are on opposite sides of the unit circle, and so the coordinates of F have the opposite signs of the coordinates of the point C, that is, F also has coordinates (sin(θ),-cos(&theta)). Therefore:

cos(θ-π/2) = sin θ & sin(θ-π/2) = - cos θ

or

cos(θ-90°) = sin θ & sin(θ-90°) = - cos θ.

Although the diagram shows the angle θ in the first quadrant, the same conclusions can be reached when θ lies in any quadrant, and so these identities hold for all angles θ.

The reason for calling these identities phase shift identities becomes clear when the functions of cos θ and sin &theta are graphed: the identities show that the graph of cos θ is the same as that of sin(&theta + π/2), which is the graph of sin &theta translated π/2 to the left. Such a translation is called a phase shift. Other identities, such as the opposite angle identities, can also be recognized as phase shift identities (with different phase shifts).

5. Opposite Angle Identities

By opposite angles we mean two angles that differ by a straight angle, that is, by π = 180°.

The angle opposite θ has measure θ + π = θ + 180°. Thus Q has coordinates
(cos(θ + π),sin(θ + π)) = (cos(θ + 180°),sin(θ + 180°)).

The point on the opposite side of the unit circle from (a,b) is the point (-a,-b). So Q also has coordinates (-cos(θ),-sin(θ)). Therefore:

cos(θ+π) = -cos θ & sin(θ+π) = -sin θ

or

cos(θ+180°) = -cos θ & sin(θ+180°) = -sin θ.

These are the opposite angle identities.

Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the opposite angle identities hold for all angles θ.

6. Negative Angle Identities (Symmetry of Sine and Cosine)

The negative -θ of an angle θ is the angle with the same magnitude but measured in the opposite direction from the positive x-axis. A positive angle θ is measured counterclockwise from the positive x-axis, so then -θ is measured clockwise from the positive x-axis.

The negative angle -θ is also the angle found by reflecting the angle θ in the x-axis. If the angle AOP is θ, then the angle AOQ is -θ. Thus Q has coordinates (cos(-θ),sin(-θ)).

When a point (a,b) is reflected in the x-axis, it moves to the point (a,-b). So Q also has coordinates (cos(θ),-sin(θ)). Therefore:

cos(-θ) = cos θ & sin(-θ) = - sin θ.

These are the negative angle identities.

Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the negative angle identities hold for all angles θ.

The negative angle identities also tell us the symmetries of the cosine and sine functions.

A function f is even if f(-x) = f(x) for every x in the domain of f. Since cos(-θ) = cos θ, we conclude that cosine is an even function. An even function y = f(x) is symmetric about the y-axis, since for each point (x,f(x)) on the graph of f, the point (-x,f(-x)) = (-x,f(x)) also lies on the graph, so for each x the y-coordinate at -x is the same as the y-coordinate at x.

A function f is odd if f(-x) = -f(x) for every x in the domain of f. Since sin(-θ) = - sin θ, we conclude that sine is an odd function. An odd function y = f(x) is symmetric about the origin, since for each point (x,f(x)) on the graph of f, the point (-x,f(-x)) = (-x,-f(x)) also lies on the graph, so for each x the y-coordinate on the graph at -x is the negative of the y-coordinate at x.


The material in this lesson is adapted from The Amazing Unit Circle at mathmistakes.info.