Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Trigonometry - finding a trig ratio given the value of another.
Test Yourself 1 - Solutions.


 

    Steps 1-3 Steps 4-6

(red tick is first bit of information).

 Steps 7-8
The 3 basic ratios 1.
  • sin θ = 3/5;
  • cos θ < 0;
  • find tan θ.
tan θ = - 3/4
  2.
  • tan α = 5/12;
  • sin α > 0;
  • find cos α.
cos α = 12/13
  3.
  • cos β = 8/17;
  • 180° ≤ β ≤ 360°;
  • find sin β.
sin β = -15/17
  4.
  • tan x = 1/3;
  • cos x < 0;
  • find sin x.
 sin x = -1/√10
  5.
  • cos x = -1/3;
  • sin x < 0;
  • find tan x.
 tan x = √8
  6.
  • sin x = -1/2;
  • tan x < 0;
  • find cos x.
 cos x = √3/2
  7.
  • sin x = -1;
  • find cos x.
  

As sin x = -1,
x = 270°.

So cos x = 0
  8.
  • cos x = 1/2;
  • tan x > 0;
  • find sin x.
 sinx = √3/2
  9.
  • tan β = -15/8;
  • 90° ≤ β ≤ 270°;
  • find cos β.
 cos β = -8/17.
  10.
  • cos θ = 5/7;
  • 0° < θ < 180°;
  • find tan θ° and
    sin θ.

tan θ = -√24/5

sin θ = √24/7
  11.
  • cos x = 5/9;
  • 180° ≤ x ≤ 360°;
  • find the value of
    sin x + tan x - cos x
    (to 2 decimal places).

sin x = -√56/9
tan x = -√56/5
cos x = 5/9

Sum = -2.88
  12.
  • tan α = -21/20;
  • cos α > 0;
  • find sin α
 sin α = -21/29
The 3 reciprocal ratios 13.
  • sec x = -2;
  • tan x < 0;
  • find sin x.
 sin x = √3/2
  14.
  • cot x = -√3;
  • sin x < 0;
  • find cos x.
 cos x = √3/2
  15.
  • cosec x = -3/√5;
  • sec x < 0;
  • find cot x.
 cot x = 2/√5
  16.
  • cot θ = 24/7;
  • sec θ < 0;
  • find cosec θ.
 cosec θ = -25/7
  17.
  • cos α = 12/13;
  • 180° < α < 360°;
  • find cot α.
cot α = -12/5
  18. cos β = -3/4
  • find sin β
    and cot β;
  • verify your answer;
  • Verify
    sin2β + cos2 β= 1.

2nd quadrant:
sin β = √7/4
cot β = -3/√7.

3rd quadrant:
sin β = -√7/4
cot β = +3/√7

  19.
  • cosec x = -5/2;
  • cos x > 0;
  • find cot x & sec x.

cot x = -√21/2

sec x = +5/√21
√(1 - x2) 20.
  • sin θ = x;
  • x > 0 and
    90° ≤ θ ≤ 270° ;
  • find tan θ and
    sec θ.
 tan θ = -x/√(1 - x2)
sec θ = -1/√(1 - x2)