Trigonometry Worksheet (Page 10 of 22) in pdf

Trigonometry Worksheet Page 10

In Quadrant 2, the cosine will be negative (since the x-coordinate of any point in Quadrant 2 is

negative). The sine will stay positive, and since rays in Quadrant 2 have negative slope, the tangent will

be negative.

In Quadrant 3, both the cosine and sine values are negative, but rays in this quadrant have positive

slope, so the tangent is positive.

In Quadrant 4, the cosine is positive, the sine is negative and the tangent is negative.

This is summarized in the following figure:

Remember, “All Students Take Calculus”. In Quadrant 1, all trigonometric functions are positive. In

Quadrant 2, the sine function is positive. In Quadrant 3, the tangent function is positive, and in

Quadrant 4, the cosine function is positive.

We use symmetry to determine the actual values for cosine, sine and tangent.

= 120°. Determine cos , sin and tan .

Example: Suppose

Solution: Since 120° is in Quadrant 2, the cosine will be negative, the sine positive and the tangent

negative. The ray that represents 120° is drawn, and we notice that it is symmetric across the y-axis with

the ray for 60°. We know from the above table that cos 60 =

, sin 60 =

and tan 60 = 3. We

attach a negative sign to the cosine and tangent values, and leave the sine value alone. Therefore,

cos 120 = −

, sin 120 =

and tan 120 = − 3.

In the above example, the 60° was the reference angle used to determine the trigonometric values for

120°. The reference angle lies in Quadrant 1, and to determine the correct reference angle for an angle

in Quadrants 2, 3 or 4, refer to the table below:

Quadrant

Reference Angle Formula

Symmetry

180 −

Across y-axis

(degrees)

−

(radians)

− 180 (degrees)

Across origin

−

(radians)

360 −

Across x-axis

(degrees)

2 −

(radians)

Trigonometry Worksheet Page 10