EVALUATING TRIG FUNCTIONS WITHOUT A CALCULATOR

In the given trigonometric function, first find the required angle lies in which quadrant.

0 ≤ θ ≤ 90 (or) 0 ≤ θ ≤ π/2

90 ≤ θ ≤ 180 (or) π/2 ≤ θ ≤ π

180 ≤ θ ≤ 270 (or) π ≤ θ ≤ 3π/2

270 ≤ θ ≤ 360 (or) 3π/2 ≤ θ ≤ 2π

θ lies in 1st quadrant

θ lies in 2nd quadrant

θ lies in 3rd quadrant

θ lies in 4th quadrant

  • It will be easy to convert the angle from radian measure to degree measure.
  • To convert the radian measure to degree measure, we have to multiply the given radian by 180/π.
  • The by drawing a special right triangle with the help of reference angle, we can easily find the exact value of the trigonometric function.

θ

180 - θ (or) π - θ

θ - 180 (or) θ - π

360 - θ (or) 2π - θ

θ lies in 1st quadrant

θ lies in 2nd quadrant

θ lies in 3rd quadrant

θ lies in 4th quadrant

  • Use ASTC to fix the signs.

Find the exact value of each trigonometric function.

Problem 1 :

sec -90°

Solution:

sec(-90°)=1cos(-90°)=10=Undefined

Problem 2 :

sin-𝜋4

Solution:

=sin-𝜋4=sin(-45°)=-22

Problem 3 :

cos 4𝜋3

Solution:

=cos 4𝜋3=cos𝜋+𝜋3=-cos𝜋3=-12

Problem 4 :

tan 45°

Solution:

=  tan 45°

= 1

Problem 5 :

csc 210°

Solution:

csc(210°)=1sin(210°)=1sin(180°+30°)=1-sin(30°)=-1sin(30°)=-112=-2

Problem 6 :

cos -3𝜋4

Solution:

=cos -3𝜋4=cos𝜋-𝜋4=-cos𝜋4=-22

Problem 7 :

sec -60°

Solution:

sec -60° = sec 60°

= sec (90° - 30°)

[sec (90 - θ) = cos θ]

= cos 30°

=1sin 30°=112=2

Problem 8 :

cot 4𝜋3

Solution:

=cot 4𝜋3=1tan4𝜋3=1tan𝜋+𝜋3=1tan𝜋3=13=33

Problem 9 :

sin 330°

Solution:

= sin 330°

= sin (360° - 30°)

= -sin(30°)

= -1/2

Problem 10 :

cot -90°

Solution:

= cot -90°

= 0

Problem 11 :

tan -3𝜋2

Solution:

=tan -3𝜋2=tan (-270°)=Undefined

Problem 12 :

sin 11𝜋6

Solution:

=sin 11𝜋6=sin2𝜋-𝜋6=-sin𝜋6=-12

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