HOW TO FIND EXACT VALUE OF TRIG FUNCTIONS WITHOUT 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.

Problem 1 :

cos -210°

Solution:

cos -210° = cos 210°

= cos (180° + 30°)

= -cos 30°

= -√3/2

Problem 2 :

cos-2𝜋3

Solution:

cos-2𝜋3=cos2𝜋3=cos𝜋-𝜋3=-cos𝜋3=-12

Problem 3 :

cot-𝜋2

Solution:

=cot-𝜋2=0

Problem 4 :

tan-𝜋2

Solution:

=tan-𝜋2=tan 90°=Undefined

Problem 5 :

csc -420°

Solution:

csc(-420°)=1sin(-420°)=-1sin 420°=-1sin(360°+60°)=-1sin 60°=-132=-23=-23×33=-233

Problem 6 :

cos 1050°

Solution:

cos 1050° = cos (330° + 720°)

= cos (330°)

= cos(360° - 30°)

= cos 30°

= √3/2

Problem 7 :

cot -840°

Solution:

cot(-840°)=1tan(-840°)=-1tan 840°=-1tan(720°+120°)=-1tan 120°=-1tan (180°-60°)=1tan 60°=13=33

Problem 8 :

csc-3𝜋4

Solution:

csc-3𝜋4=1sin-3𝜋4=-1sin𝜋-𝜋4=-1sin𝜋4=-112=-2

Problem 9 :

sin -450°

Solution:

sin -450° = - sin(360° + 90°)

= - sin 90°

= -1

Problem 10 :

sin 240°

Solution:

sin 240° = sin (180° + 60°)

= -sin 60°

= -√3/2

Problem 11 :

csc π

Solution:

csc 𝜋=1sin 𝜋=10=Undefined

Problem 12 :

csc-31𝜋6

Solution:

csc-31𝜋6=1sin-31𝜋6=1-sin31𝜋6=-1sin31𝜋6-4𝜋=-1sin7𝜋6=-1-12=2

Problem 13 :

sec -390°

Solution:

sec(-390°)=1cos(390°)=1cos(360°+30°)=1cos 30°=132=23=23×33=233

Problem 14 :

cos -930°

Solution:

cos(-930°)=cos 930°=cos(900°+30°)=cos5𝜋+𝜋6=-cos𝜋6=-32

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