Relationship Between Trigonometric Ratios

If you know a value for one trig ratio you can often use this to work out the value for the others without needing to find θ
If you know that $\sin θ = \frac{a}{b}$ , where $a, b \in ℤ^{+}$ , you can:
- Sketch a right-triangle with a opposite θ and b on the hypotenuse
- Use Pythagoras’ theorem to find the value of the adjacent side
- Use SOHCAHTOA to find the values of cos θ and tan θ
If you know a value for sin θ or cos θ you can use the Pythagorean relationship
- $\sin^{2} θ + \cos^{2} θ = 1$
- to find the value of the other
If you know a value for sin θ or cos θ you can use the double angle formulae to find the value of sin 2θ or cos 2θ
If you know a value for tan θ you can use the double angle formulae to find the value of tan 2θ
If you know two out of the three values for sin θ, cos θ or tan θ you can use the identity in tan
- $\tan θ = \frac{\sin θ}{\cos θ}$
- to find the value of the third ratio

It is possible to determine whether a trigonometric ratio will be positive or negative by looking at the size of the angle and considering the unit circle
- Angles in the range 0° < θ° < 90° will be positive for all three ratios
- Angles in the range 90° < θ° < 180° will be positive for sin and negative for cos and tan
- Angles in the range 180° < θ° < 270° will be positive for tan and negative for sin and cos
- Angles in the range 270° < θ° < 360° will be positive for cos and negative for sin and tan
The ratios for angles of 0°, 90°, 180°, 270° and 360° are either 0, 1, -1 or undefined
- You should know these ratios or know how to derive them without a calculator

Being able to sketch out the unit circle and remembering CAST can help you to find all solutions to a problem in an exam question

The value of $\sin α = \frac{3}{5}$ for $\frac{π}{2} \leq α \leq π$ . Find:

\cos α

aa-sl-3-6-3-relationship-trig-ratios-part-i-we-solution

ii)

\tan α

aa-sl-3-6-3-relationship-trig-ratios-part-ii-we-solution

iii)

\sin 2 α

aa-sl-3-6-3-relationship-trig-ratios-part-iii-we-solution

iv)

\cos 2 α

aa-sl-3-6-3-relationship-trig-ratios-part-iv-we-solution

\tan 2 α

aa-sl-3-6-3-relationship-trig-ratios-part-v-we-solution

Relationship Between Trigonometric Ratios (DP IB Maths: AA HL)