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secondary 3 | A Maths

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Since cos A = 1/2, it means that cos A has a positive value. From this information alone, we can identify that A lies in the first quadrant or the fourth quadrant.

Since sin A has the same sign as cos A, sin A has a positive value. From this information, we can identify that A lies in the first quadrant or the second quadrant.

Since A refers to the same angle, A must be in the first quadrant.

We proceed to draw a right angle triangle in the first quadrant starting at the origin and touching the x-axis. In other words, you should be drawing a diagonal line from the origin in the first quadrant to a certain point and then draw a line vertically downwards from that point.

Now, the angle touching the x-axis inside the triangle which is not 90 degrees is angle A. Since cos A = 1/2 and we know very well that cos is adj/hyp, we can set the adj length to 1 and the hyp length to 2. Solving for the opp length gives sqrt 3.

We see from here that sin A = (sqrt 3) / 2 and tan A = sqrt 3.

From here, we can proceed to solve the next part.

Since A lies in the first quadrant, -A lies in the fourth quadrant and sin in the fourth quadrant is negative,

sin (-A) = - sin A = - (sqrt 3) / 2

(the negative sign in front of sin A reflects the fact that sin in the fourth quadrant is the negative of the sin in the first quadrant).

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