Definition: The smallest angle that the
terminal side of a given angle makes with the x-axis.

Try this:
Adjust the angle below by dragging the orange point around the
origin, and note the blue reference angle.

In the figure above, as you drag the orange point around the origin, you can see the blue reference angle being drawn. It is the angle between the terminal side and the x axis. As the point moves into each quadrant, note how the reference angle is always the smallest angle between the terminal side and the x axis.

Regardless of which quadrant we are in, the reference angle is always made positive. Drag the point clockwise to make negative angles, and note how the reference angle remains positive.

As you can see from the figure above, the reference angle is always less than or equal to 90°, even for very large angles. Drag the point around the origin several times. Note how the reference angle always remain less than or equal to 90°, even for large angles.

- If necessary, first "unwind" the angle: Keep subtracting 360 from it until it is lies between 0 and 360°.
(For negative angles
*add*360 instead). - Sketch the angle to see which quadrant it is in.
2 1 3 4 - Depending on the quadrant, find the reference angle:
Quadrant Reference angle for θ 1 Same as θ 2 180 - θ 3 θ - 180 4 360 - θ

In trigonometry we use the functions of angles like sin, cos and tan. It turns out that angles that have the same reference angles always have the same trig function values (the sign may vary). So for example

sin(45) = 0.707

The angle 135° has a reference angle of 45°, so itssin(135) = 0.707

This comes in handy because we only then need to memorize the trig function values of the angles less than 90°. The rest we can find by first finding the reference angle.

Also, when solving trigonometric equations we may notice one term,such as *sin(x)* and another, *sin(π-x)*,
and realize they are going to be equal, because the second is the reference angle of the first.

- In the figure above, click 'reset' and 'hide details'.
- Drag the orange dot around the origin to a new location.
- Calculate the reference angle for it
- Click 'show details' to check your answer.

- Angle definition, properties of angles
- Standard position on an angle
- Initial side of an angle
- Terminal side of an angle
- Quadrantal angles
- Coterminal angles
- Reference angle

- Introduction to the six trig functions
- Functions of large and negative angles
- Inverse trig functions
- SOH CAH TOA memory aid
- Sine function (sin) in right triangles
- Inverse sine function (arcsin)
- Graphing the sine function
- Sine waves
- Cosine function (cos) in right triangles
- Inverse cosine function (arccos)
- Graphing the cosine function
- Tangent function (tan) in right triangles
- Inverse tangent function (arctan)
- Graphing the tangent function
- Cotangent function cot (in right triangles)
- Secant function sec (in right triangles)
- Cosecant function csc (in right triangles)

- The general approach
- Finding slant distance along a slope or ramp
- Finding the angle of a slope or ramp

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