Example #2

Sketch an angle of –45 in standard position. Then find the exact values of the cosine and the sine of –45 using a right triangle.

 Sketch the angle of measure –45 in standard position.

 

Step 2

 

Step 3

Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x–axis. The other leg is parallel to the y–axis.

The triangle contains angles of 45, 45, and 90.

 

Step 4

Find the hypotenuse of the triangle.

The hypotenuse is a radius of the unit circle.

hypotenuse = 1

Step 5

The triangle is a right isosceles triangle. Its two legs are equal.

Step 6

Find each leg of the triangle.

Each leg of the triangle is 1/(√ 2) times the hypotenuse.

Each leg

Step 7

Since the point lies in quadrant IV, its x–coordinate is positive and y–coordinate negative.

Step 8

Find the coordinates.

cos (–45) = (√ 2)/2

sin (–45) = –(√ 2)/2

The coordinates are

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