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

Relationship between trigonometric ratios

What is the relationship between sine and cosine?

• $\sin \theta =\cos \left(90-\theta \right)$ and $\cos \theta =\sin \left(90-\theta \right)$

• You can see this using a right-angled triangle with hypotenuse of length 1

  • The two other angles can be labelled $\theta$ and $\left(90-\theta \right)$

  • The sides can be found using trigonometry

How do I find the exact value of a trig ratio given a different one?

• If you know the exact value of one trig ratio

  • then you can find the exact values of the other ratios

• One method involves drawing a right-angled triangle and using SOHCAHTOA

  • For example, given that $\sin \theta =\frac{a}{b}$

    • Label the hypotenuse $b$

    • Label the opposite side $a$

    • Find the adjacent side using Pythagoras' theorem

    • Find $\cos \theta$ and $\tan \theta$ using SOHCAHTOA

    • Check whether the value should be positive or negative

• Another method involves using trigonometric identities

  • For example, given that $\sin \theta =\frac{a}{b}$

    • Use ${\sin }^2\theta + {\cos }^2\theta =1$ to find $\cos \theta$

    • Use $\tan \theta =\frac{\sin \theta }{\cos \theta }$ to find $\tan \theta$

    • Check whether the value should be positive or negative

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

• You can determine whether a trigonometric ratio is positive or negative by looking at the size of the angle

  • You can use the unit circle or the graphs