Angles and Quadrants
 

From the quadrant diagram above.

1st quadrant is from 0 to 90°, on this quadrant Sin, Cos and Tan are all positive.

2nd quadrant is 90 to 180°, only Sin is positive here while others are negative.

3rd quadrant is 180° to 270°, only tan is positive here.

4th quadrant is 270° to 360°, only cos is positive.

 
 
 

The diagram above shows that angles are measured with respect to the x – axis.

** 1st quad – angle remains the same as it is.

** 2nd quad – subtract the given angle from 180°.

** 3rd – add the given angle to 180°

** 4th – subtract from 360°.

Confused? Relax, you will see the application of all these in the examples below.

 

           Example

 

Find the values of θ between 0° and 360° for which

      1.      Sin θ = 0.345

 

Note:

      1.   0° ≤ x ≤ 360° is same as between 0° and 360°.

      2.   While 0° ≤ x ≤ 180° means between 0° and 180°. Values of x greater than 180 will be discarded here.

 

Solution

1.   sin θ = 0.345

θ = sin1 0.345

Pressing shift + sin 0.345 or 2nd function + sin 0.345 on our calculator gives

θ = 20.182°.

 

Take a look at the question again

 

sin θ = 0.345  

 

0.345 is a positive number, which means sin is positive in that quadrant. In which quadrant is sin positive? 1st and 2nd quadrant. i.e. θ and 180 – θ.

 

 

** θ = 20.182°

 

** 180 – θ

      = 180 – 20.182

       = 159.818°.

 

A quick check. Pick your calculator and press sin 20.182° and sin 159.818°.

 

 

What did you observe? The same answer.

 

∴ if sin θ = 0.345, then θ = 20.182° or 159.818°. 

 

    2.   Find the values of P between 0 and 360° for which

tan P = 1.28

P = tan–1 1.28

P = 52.00°

since tan P is positive at the 1st (P) and 3rd quad (180 + P)

= 180 + 52

= 232°.

∴ P = 52° or 232°.

 

      3.   Find the values of A between 0 and 360° for which

cos A = – 0.8329

Ignore the – sign. It only means cos is negative in that quadrant.

cos A = 0.8329

A = cos – 1 0.8329

A = 33.6°

Since Cos A is negative, then A belongs to 2nd and 3rd quadrants.

2nd quadrant = 180 – A

= 180 – 33.6

= 146.4°. 

 

3rd quadrant = 180 + A

= 180 + 33.6

= 213.6°.

∴ A = 146.4° and 213.6°.

 

You can confirm the above answers on your calculator by pressing cos 146.4° and cos 213.6°. you will get – 0.8329 in both cases.

 

       4. Find the values of θ between 0 and 360° for which

sin θ = – 0.749

sin θ = 0.749

θ = sin – 1 0.749

θ = 48.5°

sin is negative in 3rd and 4th quadrants. 180 + θ and 360 – θ.

180 + 48.5 = 228.5

360 – 48.5 = 311.5

∴ θ = 228.5° and 311.5°

 

         Exercise C 

Find the values of P between 0 and 360 for which

      1. Cos P = 0.377

      2. Tan P = – 1.5527

      3. Sin P = 0.982

      4. Cos P = – 0.848

      5. Tan P = 4

      6. Sin P = – 0.7

Find the values between 0 and 360 of the angles given below.

      7. Tan y = 1.22

      8. Cos k = 0.332

      9. Sin t = - 0.08

     10. Cos q = - 0.0952

 

                Example 2 

      1.  If sin β = - 0.4629 and cos β is negative, what quadrant does β belongs, hence find β.

 

Solution 

From our question above, sin and cos are negative. That is second quadrant.

Sin β = 0.4629

β = sin-1 0.4629

β = 25.574°.

sin and cos are negative in 2nd quad. only, i.e. 180 + β

= 180 + 25.574

= 205.574°.

∴ β = 205.574°.

 

       2.   Find t if tan t = - 2.6 and sin t is negative.

Solution

Tan t = 2.6

t = tan-1 2.6

t = 68.963°. since both tan and sin are –ve, then it is 4th quadrant.

= 360 – t

= 360 – 68.963

= 291.037°.

∴ t = 291.037°.

 

       3.   If cos q = 0.4882 and tan q is positive, find q

Solution

cos q = 0.4882

q = cos-1 0.4882

q = 60.778°.

both cos and tan are positive only on the 1st quadrant, on this quadrant, our angle remains as it is.

∴ q = 60.778°.

 

                       Exercise D

Find the angles below given the conditions stated in front.

      1. A, if cos A = -0.588 and sin A is positive

      2. B, if tan B = 1.7592 and cos B is negative

      3. C, if tan C is negative and sin C = -0.6

      4. D, if cos B = -0.83 and tan B is negative.

      5. K, if tan K = 0.69 and sin K is positive. 

 

PRE: INTRO CONTINUATION

NEXT: COMPLEMENTARY ANGLES

 

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