Relationship Between Trigonometric Ratios (DP IB Analysis & Approaches (AA)) : Revision Note
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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
, 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
Examiner Tips and Tricks
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 for
. Find:
i)

ii)

iii)

iv)

v)

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