Question

Find the angle measure that makes the statement true.
Sin 25° = Cos ____ °

Answer 1

\[\cos \left( 90-\theta) \right)=\sin \theta\]

Answer 2

Look at the figure above.

What is \(\sin A\)?

\(\sin A = \dfrac{opp}{hyp}\)

\(\sin A = \dfrac{y}{h} \)

Now again using the figure above, what is \(\cos B\)?

\(\cos B = \dfrac{adj}{hyp} \)

\(\cos B = \dfrac{y}{h}\)

In a right triangle with acute angles A and B, we have

\(\sin A = \cos B\)

Now notice what is the relation between the acute angles of a right triangle.
The measures of the three angles of any triangle add up to 180 degrees.
Since a right triangle has measure 90 degrees, that means the two acute angles of a right triangle have measures that add up to 90 degrees.
That means that A + B = 90, and B = 90 - A.

From this we derive an important relation in trigonometry.
This is known as a co-function identity.

\(\sin A = \cos (90^\circ - A)\)
\(\cos A = \sin (90^\circ - A)\)

This applies not only to sine and cosine, but to the other trig co-functions:
tangent and cotangent
secant and cosecant