Reference Triangles

A reference triangle is formed by "dropping" a perpendicular from the terminal ray of a standard position angle to the x-axis. Remember, it must be drawn to the x-axis.

Reference triangles are used to find trigonometric values for their standard position angles. They are of particular importance for standard position angles whose terminal sides reside in quadrants II, III and IV. A reference triangle contains a reference angle.

As long as the angle is in the first quadrant, it is a simple matter to find the trigonometric values associated with the angle. Piece of cake!!

To form the reference triangle, simply drop a perpendicular from the terminal ray of the angle to the x-axis, forming a right triangle.

Since the standard position angle in this example is 45º, label the triangle with the values for the 45º-45º-90º patterns
(see Special Right Triangles).

You are now ready to find all six trigonometric functions. Notice that x and y values are positive in quadrant I.

When the terminal ray moves our angle into quadrants II, III, or IV, finding the trigonometric values becomes more of a challenge. When our angle moves out of the first quadrant, we must be aware of the sign of our function value.

The reference angle and the reference triangle are used to find the trigonometric function values of angles in Quadrants II, III and IV.

Be sure to mark the signs on the legs of the right triangle. The hypotenuse is a directed segment and is considered positive.

Example 1: Find the exact value of sin 135º.

Solution: Draw the angle in standard position (with initial ray on the x-axis and opening counterclockwise). Find the reference angle (in this case 45º). Draw the reference triangle. Label the sides of the triangle with the patterns for a 45º- 45º- 90º triangle, being careful to include the appropriate sign. Now read your answer of sine from the triangle.
Answer:

Example 2: Find the exact value of csc 300º.

Solution: Draw the angle in standard position (with initial ray on the x-axis and opening counterclockwise). Find the reference angle (in this case 60º). Draw the reference triangle. Label the sides of the triangle with the patterns for a 30º- 60º- 90º triangle, being careful to include the appropriate sign. Now read your answer of cosecant from the triangle.
Answer:

Chart for signs of trigonometric function values:

Memorizing the chart at the right is not necessary as you can make these determinations by examining each quadrant.

If, however, you wish to remember such a chart, a mnemonic statement may be helpful for remembering the positive trig values (and their reciprocals) in each quadrant.

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Quadrantal Angles:
A quadrantal angle has its terminal side coinciding with a coordinate axis. The trigonometric function value of such an angle can be determined by the coordinates of the point where the unit circle intersects the axis.

	Coordinate	sin	csc	cos	sec	tan	cot
0, 0º,	(1,0)	0	undefined	1	1	0	undefined
90º,	(0,1)	1	1	0	undefined	undefined	0
180º,	(-1,0)	0	undefined	-1	-1	0	undefined
270º,	(0,-1)	-1	-1	0	undefined	undefined	0

(Again, there is no need to memorize such a chart. Simply look at the point of interest and remember that sine is vertical length and cosine is horizontal length in a unit circle. The chart is displayed here to show the combination of answers.)