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Popular >

solvefor t,sin(t)+cos(2t)=0.75

Solution

solve for

t, sin (t) + cos (2 t) = 0.75

Solution

t = - 0.18405 \dots + 2 πn, t = π + 0.18405 \dots + 2 πn, t = 0.75187 \dots + 2 πn, t = π - 0.75187 \dots + 2 πn

Degrees

t = - 10.54529 \dots^{\circ} + 36 0^{\circ} n, t = 190.54529 \dots^{\circ} + 36 0^{\circ} n, t = 43.07951 \dots^{\circ} + 36 0^{\circ} n, t = 136.92048 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (t) + cos (2 t) = 0.75

Subtract

0.75

from both sides

sin (t) + cos (2 t) - 0.75 = 0

Rewrite using trig identities

- 0.75 + cos (2 t) + sin (t)

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= - 0.75 + 1 - 2 sin^{2} (t) + sin (t)

Simplify

= sin (t) - 2 sin^{2} (t) + 0.25

0.25 + sin (t) - 2 sin^{2} (t) = 0

Solve by substitution

0.25 + sin (t) - 2 sin^{2} (t) = 0

Let:

sin (t) = u

0.25 + u - 2 u^{2} = 0

0.25 + u - 2 u^{2} = 0 : u = - \frac{- 1 + 3}{4}, u = \frac{1 + 3}{4}

0.25 + u - 2 u^{2} = 0

Multiply both sides by

100

0.25 + u - 2 u^{2} = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.25 \cdot 100 + u \cdot 100 - 2 u^{2} \cdot 100 = 0 \cdot 100

Refine

25 + 100 u - 200 u^{2} = 0

25 + 100 u - 200 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 200 u^{2} + 100 u + 25 = 0

Solve with the quadratic formula

- 200 u^{2} + 100 u + 25 = 0

Quadratic Equation Formula:

For

a = - 200, b = 100, c = 25

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 ( - 200 ) \cdot 25}{2 ( - 200 )}

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 ( - 200 ) \cdot 25}{2 ( - 200 )}

10 0^{2} - 4 (- 200) \cdot 25 = 1003

10 0^{2} - 4 (- 200) \cdot 25

Apply rule

- (- a) = a

= 10 0^{2} + 4 \cdot 200 \cdot 25

Multiply the numbers:

4 \cdot 200 \cdot 25 = 20000

= 10 0^{2} + 20000

10 0^{2} = 10000

= 10000 + 20000

Add the numbers:

10000 + 20000 = 30000

= 30000

Prime factorization of

30000 : 2^{4} \cdot 3 \cdot 5^{4}

30000

30000

divides by

230000 = 15000 \cdot 2

= 2 \cdot 15000

15000

divides by

215000 = 7500 \cdot 2

= 2 \cdot 2 \cdot 7500

7500

divides by

27500 = 3750 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3750

3750

divides by

23750 = 1875 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 1875

1875

divides by

31875 = 625 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 625

625

divides by

5625 = 125 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5

= 2^{4} \cdot 3 \cdot 5^{4}

= 2^{4} \cdot 5^{4} \cdot 3

Apply radical rule:

= 3 2^{4} 5^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 3 5^{4}

Apply radical rule:

5^{4} = 5^{\frac{4}{2}} = 5^{2}

= 2^{2} \cdot 5^{2} 3

Refine

= 1003

u_{1, 2} = \frac{- 100 \pm 100 3}{2 ( - 200 )}

Separate the solutions

u_{1} = \frac{- 100 + 100 3}{2 ( - 200 )}, u_{2} = \frac{- 100 - 100 3}{2 ( - 200 )}

u = \frac{- 100 + 100 3}{2 ( - 200 )} : - \frac{- 1 + 3}{4}

\frac{- 100 + 100 3}{2 ( - 200 )}

Remove parentheses:

(- a) = - a

= \frac{- 100 + 100 3}{- 2 \cdot 200}

Multiply the numbers:

2 \cdot 200 = 400

= \frac{- 100 + 100 3}{- 400}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 100 + 100 3}{400}

Cancel

\frac{- 100 + 100 3}{400} : \frac{3 - 1}{4}

\frac{- 100 + 100 3}{400}

Factor

- 100 + 1003 : 100 (- 1 + 3)

- 100 + 1003

Rewrite as

= - 100 \cdot 1 + 1003

Factor out common term

100

= 100 (- 1 + 3)

= \frac{100 ( - 1 + 3 )}{400}

Cancel the common factor:

100

= \frac{- 1 + 3}{4}

= - \frac{3 - 1}{4}

= - \frac{- 1 + 3}{4}

u = \frac{- 100 - 100 3}{2 ( - 200 )} : \frac{1 + 3}{4}

\frac{- 100 - 100 3}{2 ( - 200 )}

Remove parentheses:

(- a) = - a

= \frac{- 100 - 100 3}{- 2 \cdot 200}

Multiply the numbers:

2 \cdot 200 = 400

= \frac{- 100 - 100 3}{- 400}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 100 - 1003 = - (100 + 1003)

= \frac{100 + 100 3}{400}

Factor

100 + 1003 : 100 (1 + 3)

100 + 1003

Rewrite as

= 100 \cdot 1 + 1003

Factor out common term

100

= 100 (1 + 3)

= \frac{100 ( 1 + 3 )}{400}

Cancel the common factor:

100

= \frac{1 + 3}{4}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 3}{4}, u = \frac{1 + 3}{4}

Substitute back

u = sin (t)

sin (t) = - \frac{- 1 + 3}{4}, sin (t) = \frac{1 + 3}{4}

sin (t) = - \frac{- 1 + 3}{4}, sin (t) = \frac{1 + 3}{4}

sin (t) = - \frac{- 1 + 3}{4} : t = arcsin (- \frac{- 1 + 3}{4}) + 2 πn, t = π + arcsin (\frac{- 1 + 3}{4}) + 2 πn

sin (t) = - \frac{- 1 + 3}{4}

Apply trig inverse properties

sin (t) = - \frac{- 1 + 3}{4}

General solutions for

sin (t) = - \frac{- 1 + 3}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

t = arcsin (- \frac{- 1 + 3}{4}) + 2 πn, t = π + arcsin (\frac{- 1 + 3}{4}) + 2 πn

t = arcsin (- \frac{- 1 + 3}{4}) + 2 πn, t = π + arcsin (\frac{- 1 + 3}{4}) + 2 πn

sin (t) = \frac{1 + 3}{4} : t = arcsin (\frac{1 + 3}{4}) + 2 πn, t = π - arcsin (\frac{1 + 3}{4}) + 2 πn

sin (t) = \frac{1 + 3}{4}

Apply trig inverse properties

sin (t) = \frac{1 + 3}{4}

General solutions for

sin (t) = \frac{1 + 3}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

t = arcsin (\frac{1 + 3}{4}) + 2 πn, t = π - arcsin (\frac{1 + 3}{4}) + 2 πn

t = arcsin (\frac{1 + 3}{4}) + 2 πn, t = π - arcsin (\frac{1 + 3}{4}) + 2 πn

Combine all the solutions

t = arcsin (- \frac{- 1 + 3}{4}) + 2 πn, t = π + arcsin (\frac{- 1 + 3}{4}) + 2 πn, t = arcsin (\frac{1 + 3}{4}) + 2 πn, t = π - arcsin (\frac{1 + 3}{4}) + 2 πn

Show solutions in decimal form

t = - 0.18405 \dots + 2 πn, t = π + 0.18405 \dots + 2 πn, t = 0.75187 \dots + 2 πn, t = π - 0.75187 \dots + 2 πn

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