TRIGONOMETRY EQUATIONS

             Example

Solve the equations below for values of y between 0° and 360° 

 

        1.   5sin y + 1 = 3 

                 Solution

 5sin y + 1 = 3

Collect like terms

5sin y = 3 – 1

5sin y = 2

Divide both sides by 5

Sin y = 2/5

Sin y = 0.4

y = sin – 1 0.4 (sin is positive)

y = 23.579°              or        y = 180 – 23.579°

y = 156.421°.

 

        2.   cos2y = 1/25

                        Solution

Cos2y = 1/25 take the square root of both sides

Cos y = ±√1/25

Cos y = ± 1/5

Cos y = ± 0.2

Split the equation

Cos y = 0.2                            or                    cos y = - 0.2

y = cos – 1 0.2                                               y = cos – 1 0.2

y = 78.463°                                                  y = 78.463°

cos is +ve                                                      cos is –ve

y = 360 – 78.463                                        y = 180 – 78.463; y = 101.537°

y = 281.537°                                                 y = 180 + 78.463; y = 258.463°

∴ y = 78.463°, 281.537°, 101.537°, 258.463°

 

        3.    6 + 25tan2x = 27

               Solution

6 + 25tan2x = 27

25tan2x = 27 – 6

25tan2x = 21

tan2x = 21/25

tan2x = 0.84

tan x = ±√0.84

tan x = ± 0.9185

tan x = 0.9185                     or                    tan x = - 0.9185

x = tan – 1 0.9185                                                                                                               

x = 42. 5057°                                                          

tan is positive                                              tan is negative

x = 180 + 42. 5057                                     x = 180 - 42.5057 or x = 360 – 42. 5057

x = 222.5057°                                             x = 137.4943°      or    x = 317.4943°

∴ x = 42. 5057°, 222.5057°, 137.4943°, 317.4943°

 

        4.    5 sin y + 2 = 6 + 11 sin y

                        Solution

5 sin y + 2 = 6 + 11 sin y

Collect like terms

5 sin y – 11 sin y = 6 – 2

– 6 sin y = 4

Sin y = – 4/6

Sin y = – 0.6667

y = sin – 1 0.6667

y = 41.8103°

sin is negative (3rd and 4th quadrant)

y = 180 + 41.8103              or                    y = 360 – 41.8103

y = 221.8103°                                              y = 318.1897°

∴ y = 221.8103°, 318.1897°.

 

            Exercise

Solve the equations below for the values of the angles between 0 and 360°

        1.   6 sin x = 5

        2.   2 + 7cos p = 5

        3.   12 tan2k = 7

        4.   5 sin x + 8 = 7

        5.   8 cos2 y = 3

        6.   7 + 15 cos x = 5

        7.   6 tan2k – 7 = 5

        8.   13 – 5 cos2 p = 9

        9.   28 = 35 – 13 sin2 θ

       10.   8 – 3 cos x = 13 – 10 cos x

       11.   6 tan2 Q – 9 = 3 – 5 tan2 Q

 

              Example 2 

Solve the equations below

        1.    2 tan2P – tan P – 10 = 0

        2.    8 sin k = 3 – 4 sin2k

            Solution

The two equations above are quadratic equations, so we solve them using the various method for solving quadratic equation.

 

    1.    2 tan2P – tan P – 10 = 0                  (see it as if you want to solve 2p2 – p – 10 = 0).

            Factorize

            2tan2p + 4tan p – 5tan p – 10 = 0

            (2tan2p + 4tan p) – (5tan p + 10) = 0

            2 tan p (tan p + 2) – 5 (tan p + 2) = 0

            (tan p + 2)( 2tan p – 5) = 0

            Tan p + 2 = 0                                                or                    2tan – 5 = 0

            Tan p = 0 – 2                                                                        2 tan p = 0 + 5

            tan p = - 2                                                                             2 tan p = 5

            p = tan – 1 2                                                                           tan p = 5/2

            p = 63.4349                                                                         tan p = 2.5

            tan is negative                                                                    p = tan – 1 2.5  (tan is +ve)

            p = 180 – 63. 4349 or 360 – 63. 4349                          p = 68.1986°           

p = 116.5651° or 296.5651°                                                      or p = 180 + 68.1986

                                                                                                            p = 248.1986°

 

∴ p = 116.5651°, 296.5651°, 68.1986°, 248.1986°.

 

 

        2.   8 sin k = 3 – 4 sin2k

            Rearrange

4 sin2k + 8 sin k – 3 = 0                                 (see it as if you want to solve 4k2 + 8 k – 3 = 0)

We cannot solve the above equation using factorization method, hence we apply quadratic formula.

a = 4, b = 8, c = - 3.

 \sin k = \frac{- b \pm \sqrt{b^2 - 4 ac}}{2a} \\ = \frac{- 8 \pm \sqrt{8^2 - 4 \times 4\times -3}}{2\times 4} \\ \\ = \frac{- 8 \pm \sqrt{64 + 48}}{8} \\ \\ = \frac{- 8 \pm \sqrt{112}}{8} \\ \\ = \frac{- 8 \pm 10.583}{8} \\ \\

sink = (- 8 + 10.583)/8                         or                           sink = (- 8 - 10.583)/8

sink = 2.583/8                                                                     sink = - 18.583/8

sink = 0.3229                                                                      sink = - 2.3229

k = sin – 10.3229                                                                  k = sin – 12.3229

k = 18.8384°                                                                        k = 󠇯ꝏ

k = 180 – 18.8384°

k = 161.1616°

 ∴ k = 18.8384°, 161.1616°

 

Exercise

1 . 12cos2y = cos y + 2

2 . 2sin2α + 3sinα = 2

3 . 5sinβ + 6 = 2 sin2β

4 . 3tan2p + 2tanp = 0

5 . 2 + coty = cot2y

6 . 0 = 3sin2α – 7sinα – 5

7 . 15cosx + 2cos2x = – 7

 

            Example 3

1 .  3cosx = cotx

2 . 5tanx = 3sinx

3 . 7secx = 12tanx

 

            Solution

1 .       3cosx = cotx

            \frac {3cosx}{1} = \frac {cosx}{sinx}

Cross multiply

            3 cosx sinx – cos x = 0

            Cosx (3sinx – 1) = 0

            Cos x = 0                                                       or                    3sinx – 1 = 0

            X = cos – 10                                                                            3sinx = 1

            x = 90°                                                                                   sinx = 1/3

                        or                                                                                sinx = 0.333     (sin is ­­­+ve)

             x = 360 – 90                                                                        x = sin – 1 0.333

            x = 270°                                                                                x = 19. 4692°

                                                                                                            or

                                                                                                            x = 180 – 19.4692

                                                                                                            x = 160.5308°

            ∴ x = 90°, 270°, 19. 4692°, 160.5308°

 

2 .       5tanx = 3sinx 

        \frac{5 sin x}{cos x} = \frac {3sinx}{1} 

Cross multiply

            3cosx sinx = 5 sinx

            3cosx sinx – 5 sinx = 0

            Sin x (3cosx – 5) = 0

Sin x = 0                                 or                                            3cosx – 5 = 0

X = sin – 1 0                                                                            3cos x = 5

x = 0                                                                                       cos x = 5/3

or                                                                                            cos x = 1.6667

x = 0 + 180                                                                            x = cos – 1 1.6667

x = 180                                                                                  x = ꝏ

 

∴ x = 0°, 180°

 

3 .       7secx = 12tanx

7(1/cosx) = 12 (sinx/cosx)

7/cosx = 12sinx/cosx

Cross multiply

12 sinx cosx  = 7 cosx

12 sinx cosx – 7 cosx = 0

cosx (12 sinx – 7) = 0

cos x = 0                                            or                                12 sinx – 7 = 0

x = cos – 1 0                                                                           12 sinx = 7

x = 90 or 360 – 90                                                              sinx = 7/12

x = 270                                                                                  sinx = 0.5833

                                                                                                x = sin – 1 0.5833

                                                                                                x = 35.683

                                                                                                x = 144.317

            ∴ x = 90°, 270°, 35.683°, 144.317°

 

Exercise

Solve the equations below for the given angles between 0 and 360

1 .       4 cosec p = 7 cot p

2 .       13 tan x = 2 sec x

3 .       3cos2x = 2 cosx

4 .       8cot2k = 11 cos2k

 

            Example 4

Solve the equations below

1 .       cos2x + sin x + 1 = 0

2 .       2 cot2x – 7 cosec x + 8 = 0

 

            Solution

1 .       cos2x + sin x + 1 = 0

From sin2x + cos2x = 1       (make cos2x the subject)

            cos2x = 1 – sin2x      (substitute 1 – sin2x for cos2x)

1 – sin2x + sin x + 1 = 0

2 – sin2x + sin x = 0

– sin2x + sin x + 2 = 0         (clear the –)

sin2x – sin x – 2 = 0

sin2x + sinx – 2sinx – 2 = 0

(sin2x + sinx) – (2sinx + 2) = 0

Sinx (sinx + 1) – 2(sin x + 1) = 0

(sinx + 1)(sin x – 2 ) = 0

sin x + 1 = 0                                                  or                                sin x – 2 = 0

sin x = – 1                                                                                         sin x = 2

x = sin – 1 – 1                                                                                     x = sin – 1 2

x = 90                                                                                                 x = ꝏ

sin is negative ( 2nd and 3rd quadrants)

x = 180 + 90 or 360 - 90

x = 270°

 

∴ x = 270°

 

2 .       2cot2x – 7cosec x + 8 = 0

            From 1 + cot2x = cosec2x              (make cot2x the subject)

            cot2x = cosec2x – 1

            2 (cosec2x – 1) – 7cosec x + 8 = 0

            2cosec2x – 2 – 7cosec x + 8 = 0

            2cosec2x – 7cosec x + 6 = 0

            2cosec2x – 4cosec x – 3cosec x + 6 = 0

            (2cosec2x – 4cosec x) – (3cosec x – 6) = 0

            2 cosec x (cosec x – 2) – 3(cosec x – 2) = 0

            (cosec x – 2)(2 cosec x – 3) = 0

cosec x – 2 = 0                                 or                                2 cosec x – 3 = 0

cosec x = 0 + 2                                                                     2 cosec x = 3

cosec x = 2                                                                            cosec x = 3/2

since you can’t press cosec directly from your calculator, change it to sin

\frac{1}{\sin x}= \frac{2}{1}                                           or                                    \frac{1}{\sin x} = \frac{3}{2}

Take the reciprocal of both sides

Sin x = ½                                            or                                sin x = 2/3

Sin x = 0.5                                                                             sin x = 0.6667

x = 30°                                                                                               x = 41.81°

x = 150°                                                                                           x = 138.19°

 

            Exercise

Solve the equations below for values of p between 0 and 360°

1 .   sec2p – 3tanp + 1 = 0

2 .   2 cot p + tan p – 3 = 0

3 .   3 sin p = 2 – 2sin2p

4 .   cot p = tan p

5 .   5 cos p = 3

6 .   cot p = 3 – tan p

7 .   sin p = 2cosp

8 .   sec2p + tanp = 3

9 .   5 cos2p + 5sinp = 1

10 .  3cosec p = 4 sin p – 4

11 . 18 cosec p = 3 – 5 cot2p

12 .  5/cotp = 2sec2p – 4

13 .  \cos p + \sin p = \frac{1}{\cos p - \sin p}

 

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