Question

sin (90° + A) sin (180° - A) tan (180° - A)
cos (360° - A) cos (180° - A) tan (180° + A)​

Answer 1

Answer:

CONTENT

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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 .

EXERCISE 1

Place the point P on the unit circle corresponding to each of the angles 0°, 90° , 180°, 270°. By considering the coordinates of each such point, complete the table below

P Θ sin θ cos θ tan θ

0°

90° undefined

180°

270° undefined

360°

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THE FOUR QUADRANTS

The coordinate axes divide the plane into four quadrants, labeled First, Second, Third and Fourth as shown. Angles in the third quadrant, for example, lie between 180° and 270°.

By considering the x and y coordinates of the point P as it lies in each of the four quadrants, we can identify the sign of each of the trigonometric ratios in a given quadrant. These are summarised in the following diagrams.

To obtain the second diagram, we used the definition tan θ = .

To assist in remembering the signs of the three trigonometric ratios in the various quadrants, we see that only one of the ratios is positive in each quadrant. Hence we can remember the signs by the picture:

The mnemonic All Stations To Central is sometimes used.

EXAMPLE

What is the sign of

a cos 150° b sin 300° c tan 235°?

SOLUTION

a 150° lies in the second quadrant so cos 150° is negative.

b 300° lies in the fourth quadrant so sin 300° is negative.

c235° lies in the third quadrant so tan 235° is positive.