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Sine and Cosine Functions

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A series of free, online Trigonometry Video Lessons.
Examples, solutions, videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

  • the radian measure of angles and how to convert from degrees to radians
  • how to define sine and cosine using the unit circle
  • how to evaluate and remember sine and cosine at special acute angles

Radian Measure of Angles

An angle is the figure formed by two rays with a common endpoint. We typically use degree measures when measuring angles, however we can use radian angle measure as an alternate way of measuring angles in advanced math courses. This measure is based on using a point on the vertex and measuring the arc length compared to the radius.

How to define radian measure?
How to convert from degrees to radians?
How to determine trigonometric function values given in radians?
Examples of converting angles in radian measure to degree measure.

The Definitions of Sine and Cosine

The right triangle definitions of sine and cosine only apply to acute angles, so a more complete definition is needed. The point where the terminal side intersects the unit circle (x, y) is the basis for this definition. Since the radius (and therefore hypotenuse of the right triangle) is 1, the denominators cosine=adjacent/hypotenuse and sine=opposite/hypotenuse are also 1. Thus, the sine definition is y=sine and x=cosine.

How we define sine and cosine for all angle measures using the unit circle?
How the unit circle is used to extend the definition of sine, cosine and tangent to angles greater than 90 degrees?
It introduces angles in all 4 quadrants, looks at how the sign of the trig functions changes in the different quadrants, how the graphs of the sine and cosine functions are related to the new definition and finally how the sine, cosine and tangent values are related to the first quadrant angle values.

Evaluating Sine and Cosine at Special Acute Angles

If you want to make your math life incredibly easier, memorize the sine and cosine values for π/3, π/4, and π/6, as well as 0, π/2, and π. These values constantly reappear throughout Trigonometry and pre-Calculus problems and proofs. When evaluating sine It is also useful to memorize the conversion from radians to degrees for these values; for example, to remember that π/6 is equivalent to 30°.

Sine and Cosine - Examples calculating the sine and cosine function for different radian values.
A Trick to Remember Values on The Unit Circle

Evaluating Sine and Cosine at Other Special Angles

To find the value of sine and cosine at non-acute angles (from 90 to 360), first draw the angle on the unit circle and find the reference angle. A reference angle is formed by the terminal side and the x-axis and will therefore always be acute. When evaluating cosine and sine for the reference angle, determine if each value is positive or negative by identifying the quadrant the terminal side is in.

How to measure the sine and cosine of 150°? Evaluating Trigonometric Functions Using the Reference Angle

Reference Angle for an Angle, Ex 1 (Using Radians).

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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