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Trigonometric Values in Quadrants

Having looked at trigonometric ratios in quadrants, we now look at examples of calculating trigonometric values in quadrants.

 

Example 1: sin 30°  =  0.5

 

Example 2: sin 100°  =  sin (180 – 80)

=  sin 80°

=  0.9848

Since 100° is in the second quadrant, the sine value is positive

 

Example 3: cos 120°  =  cos (180 – 60)

=  -cos(60°)  =  -0.5

using cos (180 – θ)  =  -cos θ

Cosine value in second quadrant is negative, as 120° is in second quadrant

 

Example 4: tan 60°  =  1.732. What is tan 120°?

tan 120°  =  tan (180 – 60)

=  -tan (60°)  =  -1.732

120° is in second quadrant, and tangent value in second quadrant is negative.

 

Example 5: sin 200°  =  sin (180 + 20)

This angle is in the third quadrant, and sine value is negative in this quadrant

sin (180 + θ)  =  -sin θ

So sin 200  =  sin (180 + 20)  =  -sin 20°

=  -0.342

 

Example 6: cos 300°  =  cos (360 – 60)

300° is in the fourth quadrant, and cosine value is positive in this quadrant

cos (360 – θ)  =  cos θ

So cos 300°  =  cos (360 – 60)

=  cos 60°

=  0.5

 

Example 7: tan (-50°)  =  tan (360 – 50)

-50° is in the fourth quadrant, and tangent value is negative in this quadrant

tan (360 – θ)  =  -tan θ  =  tan (-θ)

So tan (-50)  =  -tan 50°

=  -1.1918

 

Example 8: tan (-110°)  =  ?

Angle -110°  =  360 – 110  =  250° which is in the third quadrant. So tangent value is positive in third quadrant.

Hence tan (-110°)  =  -tan (110°)  =  -tan (180° – 70°)

= −(−tan 70°)  =  tan (70°)

=  2.7475

 

Trigonometric Values without Calculator

Without using the calculator, let us try to answer whether these trigonometric values are positive or negative.

1.  cos 280°  =  ?

280° lies in the fourth quadrant, so cosine is positive

2.  sin 300°  =  ?

300° lies in the fourth quadrant, and sine is negative

3.  tan 240°  =  ?

240° lies in the third quadrant, and tan is positive

4.  sin 210°  =  ?

210° lies in the third quadrant, and sine is negative

5.  cos 170°  =  ?

170° lies in the second quadrant, and cosine is negative