Trigonometry and the Unit Circle

Learning Objectives

Define the three primary trigonometric functions (sine, cosine, tangent) in the context of a right triangle.

Define sine and cosine in the context of the unit circle.

Convert between degrees and radians.

Identify the values of sine and cosine for key angles (0, 30, 45, 60, 90 degrees).

Measuring Triangles and Circles

Trigonometry is the branch of mathematics concerned with the relationships between the angles and side lengths of triangles.

Right Triangle Trigonometry

For an acute angle θ in a right triangle:

SOH CAH TOA is a fundamental mnemonic.

Sine(θ) = Opposite / Hypotenuse (SOH)

Cosine(θ) = Adjacent / Hypotenuse (CAH)

Tangent(θ) = Opposite / Adjacent (TOA)

The Unit Circle

A more powerful definition comes from the unit circle, which is a circle with a radius of 1 centered at the origin of the Cartesian plane.

For any angle θ measured counterclockwise from the positive x-axis, the point (x, y) where the angle's terminal side intersects the unit circle has coordinates:

x = cos(θ)

y = sin(θ)

This definition works for any angle, not just acute angles in a right triangle.

From this, we also get tan(θ) = y/x = sin(θ)/cos(θ).

Degrees vs. Radians

Degrees: A full circle is 360°.

Radians: A measure of angle based on the radius of a circle. A full circle is 2π radians. The arc length subtended by an angle in radians is s = rθ.

Conversion:

To convert from degrees to radians, multiply by π / 180.

To convert from radians to degrees, multiply by 180 / π.

Key Angles on the Unit Circle

It is crucial to know the (cosine, sine) coordinates for key angles:

0° (0 rad): (1, 0)

30° (π/6 rad): (√3/2, 1/2)

45° (π/4 rad): (√2/2, √2/2)

60° (π/3 rad): (1/2, √3/2)

90° (π/2 rad): (0, 1)

The signs of sine and cosine in the four quadrants can be remembered with 'All Students Take Calculus':

Quadrant I: All are positive.

Quadrant II: Sine is positive.

Quadrant III: Tangent is positive.

Quadrant IV: Cosine is positive.

Key Terms

Unit Circle: A circle with a radius of one, centered at the origin, which is used to define the trigonometric functions sine and cosine for all real numbers.
SOH CAH TOA: A mnemonic device for remembering the definitions of the three primary trigonometric functions: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Radian: The standard unit of angular measure, defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius.
Sine: In a right-angled triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse. On the unit circle, it is the y-coordinate.
Cosine: In a right-angled triangle, the ratio of the length of the adjacent side to the length of the hypotenuse. On the unit circle, it is the x-coordinate.