Question

10 Given that tan @ =
and is an acute angle, find sin 0 and cos 0.
​

Answer 1

Answer:

ANGLES IN THE FOUR QUADRANTS

Redefining the Trigonometric Ratios

We begin by taking the circle of radius 1, centre the origin, in the plane. From the point P on the circle in the first quadrant we can construct a right-angled triangle POQ with O at the origin and Q on the x-axis.

We mark the angle POQ as θ.

Since the length OQ = cos θ is the x-coordinate of P, and PQ = sin θ is the y-coordinate of P, we see that the point P has coordinates

(cos θ, sin θ).

We measure angles anticlockwise from OA and call these positive angles. Angles measured clockwise from OA are called negative angles. For the time being we will concentrate on positive angles between 0° and 360°.

Since each angle θ determines a point P on the unit circle, we will define

the cosine of θ to be the x-coordinate of the point P

the sine of θ to be the y-coordinate of the point P.

For acute angles, we know that tan θ = . For angles that are greater than 90° we define the tangent of θ by

tan θ = ,

unless cos In this case, we say that the tangent ratio is undefined. Between 0° and 360°, this will happen when θ = 90°, or θ = 270°. You will see in the following exercise why this is the case.

Note that this is the same as saying that tan θ equals the