Question

Can you please explain to me how to do this please

Answer 1

Explanation:

You asked how to do it, so that is the answer we will give. (The question would be "too complex" if you asked for answers to all 18 questions.)

First of all, recognize that angle values are given in degrees for some problems* and radians for other problems. Know that π radians is 180°, so you can convert to degrees by replacing π with 180°.

1. Find the quadrant of the angle:

  0 to 90° is Quadrant I

  90° to 180° is Quadrant II

  180° to 270° is Quadrant III

  270° to 360° is Quadrant IV

The signs of the trig functions in the different quadrants are ...

  sine -- positive in I and II, negative in III and IV

  cosine -- positive in I and IV, negative in II and III

  tangent -- positive in I and III, negative in II and IV

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2. Find the reference angle. The reference angle for angle α is the smallest of ...

  |α| or |180° -α| or |360° -α|

It will be a positive number in the range 0° to 90°.

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3. Make use of the short table of trig function values you have memorized. This gives you the exact value of the reference angle you found in step 2.

  sin(0°) = cos(90°) = 0

  sin(30°) = cos(60°) = 1/2

  sin(45°) = cos(45°) = (√2)/2

  sin(60°) = cos(30°) = (√3)/2

  sin(90°) = cos(0°) = 1

As always, the tangent is the ratio of sine to cosine, so you have ...

  tan(0°) = 0

  tan(30°) = (√3)/3

  tan(45°) = 1

  tan(60°) = √3

  tan(90°) = undefined

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4. Apply the sign of the desired function in the desired quadrant to the value you found in step 3. (For non-zero function values, the sign on a quadrant boundary matches the signs for the quadrants on either side.)

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Examples:

• cos(225°) = -cos(45°) = -(√2)/2 . . . . (quadrant III, ref angle 45°)
• sec(270°) = 1/cos(90°) = 1/0 = undefined . . . . (ref angle 90°)
• cot(5π/6) = cot(150°) = 1/-tan(30°) = -√3 . . . . (quadrant II, ref angle 30°)

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* Technically, sin 60 should be interpreted as sin(60 radians), since there is no degree symbol present. In this context, we can reasonably assume that values not a multiple of pi will be in degrees. (That may not always be the case.) You should always be careful to specify what unit of measure is being used for angles--even if your curriculum materials are not so careful. Your calculator is very particular on that point.