IBMaths: AA HLDPRevision Notes3. Geometry & Trigonometry3.6 Trigonometric Equations & Identities3.6.4 Relationship Between Trigonometric Ratios

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

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Relationship Between Trigonometric Ratios

What relationships between trigonometric ratios should I know?

  • 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 space theta equals a over b , where a comma space b space element of space straight integer numbers to the power of plus, 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 to the power of 2 space end exponent theta plus blank cos to the power of 2 space end exponent theta equals 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 space theta equals fraction numerator sin space theta over denominator cos space theta end fraction
    • to find the value of the third ratio

How do we determine whether a trigonometric ratio will be positive or negative?

  • 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

Exam Tip

  • 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 

Worked example

The value of sin space alpha equals 3 over 5 for straight pi over 2 space less or equal than space alpha space less or equal than space straight pi.  Find:

i)
cos space alpha

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

ii)
tan space alpha

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

iii)
sin space 2 alpha

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

iv)
cos space 2 alpha

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

v)
tan space 2 alpha

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

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Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.