Question

1/sin(√3/2)​

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5 days 2021-09-14T00:18:27+00:00 1 Answer 0 views 0

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.

The related angle

To find the trigonometric ratio of angles beyond 90°, we introduce the notion of the related angle. We will examine this quadrant by quadrant.

Suppose we wish to find the exact°and sin 150°.

The angle 150° corresponds

to the point P in the second quadrant with coordinates

(cos 150°, sin 150°) as shown.

The angle POQ is 30° and is called the related angle for 150°. From trianglePOQ we can see that OQ = cos 30° and PQ = sin 30°, so the coordinates of P are −cos 30°, sin 30°).

Hence  cos 150° = −cos 30° = −D5b211.pdf and sin 150° = sin 30° = .

Also,   tan 150° = D5b233.pdf = −tan 30° = −.

In general, if θ lies in the second quadrant, the acute angle 180° − θ is called the related angle for θ.

We introduced this idea in the module, Further Trigonometry.

An angle between 180° and 270°

will place the corresponding point P in the third quadrant. In this quadrant, we can see that the sine and cosine ratios are negative and the tangent ratio positive.

To find the sine and cosine of 210° we locate the corresponding point P in the third quadrant. The coordinates of P are (cos 210°, sin 210°). The angle POQ is 30° and is called the related angle for 210°.

So,  cos 210° = −cos 30° = −D5b315.pdf and sin 210° = −sin 30° = −.

Hence   tan 210° = tan 30° = −.

In general, if lies in the third quadrant, the acute angle θ − 180° is called the related angle for θ.

Finally, if θ lies between 270° and 360° the corresponding point P is in the fourth quadrant. In this quadrant, we can see that the sine and tangent ratios are negative and the cosine ratio is positive.

To find the sine and cosine of 330° we locate the corresponding point P in the fourth quadrant. The coordinates of P are (cos 330°, sin 330°) .The angle POQ is 30° and is called the related angle for 330°.

So,  cos 330° = cos 30° = D5b418.pdf and sin 330° = −sin 30° = −.

Hence,   tan 330° = −tan 30° = −.

In general, if lies in the fourth quadrant, the acute angle is called the related angle for θ.

In summary, to find the trigonometric ratio of an angle between 0° and 360° we

find the related angle,

obtain the sign of the ratio by noting the quadrant,

evaluate the trigonometric ratio of the related angle and attach the appropriate sign.

Step-by-step explanation: