Relationship Between Trigonometric Ratios (DP IB Analysis & Approaches (AA)): Revision Note

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 , where , 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

  • 

  • 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 two out of the three values for sin θ, cos θ or tan θ you can use the identity in tan

  • 

  • 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