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Popular Trigonometry >

15cos^2(θ)+2sin^2(θ)=7

Solution

15 cos^{2} (θ) + 2 sin^{2} (θ) = 7

Solution

θ = 0.90183 \dots + 2 πn, θ = π - 0.90183 \dots + 2 πn, θ = - 0.90183 \dots + 2 πn, θ = π + 0.90183 \dots + 2 πn

Degrees

θ = 51.67118 \dots^{\circ} + 36 0^{\circ} n, θ = 128.32881 \dots^{\circ} + 36 0^{\circ} n, θ = - 51.67118 \dots^{\circ} + 36 0^{\circ} n, θ = 231.67118 \dots^{\circ} + 36 0^{\circ} n

Solution steps

15 cos^{2} (θ) + 2 sin^{2} (θ) = 7

Subtract

7

from both sides

15 cos^{2} (θ) + 2 sin^{2} (θ) - 7 = 0

Rewrite using trig identities

- 7 + 15 cos^{2} (θ) + 2 sin^{2} (θ)

Use the Pythagorean identity:

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

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

= - 7 + 15 (1 - sin^{2} (θ)) + 2 sin^{2} (θ)

Simplify

- 7 + 15 (1 - sin^{2} (θ)) + 2 sin^{2} (θ) : - 13 sin^{2} (θ) + 8

- 7 + 15 (1 - sin^{2} (θ)) + 2 sin^{2} (θ)

Expand

15 (1 - sin^{2} (θ)) : 15 - 15 sin^{2} (θ)

15 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 15, b = 1, c = sin^{2} (θ)

= 15 \cdot 1 - 15 sin^{2} (θ)

Multiply the numbers:

15 \cdot 1 = 15

= 15 - 15 sin^{2} (θ)

= - 7 + 15 - 15 sin^{2} (θ) + 2 sin^{2} (θ)

Simplify

- 7 + 15 - 15 sin^{2} (θ) + 2 sin^{2} (θ) : - 13 sin^{2} (θ) + 8

- 7 + 15 - 15 sin^{2} (θ) + 2 sin^{2} (θ)

Add similar elements:

- 15 sin^{2} (θ) + 2 sin^{2} (θ) = - 13 sin^{2} (θ)

= - 7 + 15 - 13 sin^{2} (θ)

Add/Subtract the numbers:

- 7 + 15 = 8

= - 13 sin^{2} (θ) + 8

= - 13 sin^{2} (θ) + 8

= - 13 sin^{2} (θ) + 8

8 - 13 sin^{2} (θ) = 0

Solve by substitution

8 - 13 sin^{2} (θ) = 0

Let:

sin (θ) = u

8 - 13 u^{2} = 0

8 - 13 u^{2} = 0 : u = \frac{2 26}{13}, u = - \frac{2 26}{13}

8 - 13 u^{2} = 0

Move

8

to the right side

8 - 13 u^{2} = 0

Subtract

8

from both sides

8 - 13 u^{2} - 8 = 0 - 8

Simplify

- 13 u^{2} = - 8

- 13 u^{2} = - 8

Divide both sides by

- 13

- 13 u^{2} = - 8

Divide both sides by

- 13

\frac{- 13 u ^{2}}{- 13} = \frac{- 8}{- 13}

Simplify

u^{2} = \frac{8}{13}

u^{2} = \frac{8}{13}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{8}{13}, u = - \frac{8}{13}

\frac{8}{13} = \frac{2 26}{13}

\frac{8}{13}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{8}{13}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{2 2}{13}

Rationalize

\frac{2 2}{13} : \frac{2 26}{13}

\frac{2 2}{13}

Multiply by the conjugate

\frac{13}{13}

= \frac{2 2 13}{13 13}

2213 = 226

2213

Apply radical rule:

a b = a \cdot b

213 = 2 \cdot 13

= 2 2 \cdot 13

Multiply the numbers:

2 \cdot 13 = 26

= 226

1313 = 13

1313

Apply radical rule:

a a = a

1313 = 13

= 13

= \frac{2 26}{13}

= \frac{2 26}{13}

- \frac{8}{13} = - \frac{2 26}{13}

- \frac{8}{13}

Simplify

\frac{8}{13} : \frac{2 2}{13}

\frac{8}{13}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{8}{13}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{2 2}{13}

= - \frac{2 2}{13}

Rationalize

- \frac{2 2}{13} : - \frac{2 26}{13}

- \frac{2 2}{13}

Multiply by the conjugate

\frac{13}{13}

= - \frac{2 2 13}{13 13}

2213 = 226

2213

Apply radical rule:

a b = a \cdot b

213 = 2 \cdot 13

= 2 2 \cdot 13

Multiply the numbers:

2 \cdot 13 = 26

= 226

1313 = 13

1313

Apply radical rule:

a a = a

1313 = 13

= 13

= - \frac{2 26}{13}

= - \frac{2 26}{13}

u = \frac{2 26}{13}, u = - \frac{2 26}{13}

Substitute back

u = sin (θ)

sin (θ) = \frac{2 26}{13}, sin (θ) = - \frac{2 26}{13}

sin (θ) = \frac{2 26}{13}, sin (θ) = - \frac{2 26}{13}

sin (θ) = \frac{2 26}{13} : θ = arcsin (\frac{2 26}{13}) + 2 πn, θ = π - arcsin (\frac{2 26}{13}) + 2 πn

sin (θ) = \frac{2 26}{13}

Apply trig inverse properties

sin (θ) = \frac{2 26}{13}

General solutions for

sin (θ) = \frac{2 26}{13}

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

θ = arcsin (\frac{2 26}{13}) + 2 πn, θ = π - arcsin (\frac{2 26}{13}) + 2 πn

θ = arcsin (\frac{2 26}{13}) + 2 πn, θ = π - arcsin (\frac{2 26}{13}) + 2 πn

sin (θ) = - \frac{2 26}{13} : θ = arcsin (- \frac{2 26}{13}) + 2 πn, θ = π + arcsin (\frac{2 26}{13}) + 2 πn

sin (θ) = - \frac{2 26}{13}

Apply trig inverse properties

sin (θ) = - \frac{2 26}{13}

General solutions for

sin (θ) = - \frac{2 26}{13}

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

θ = arcsin (- \frac{2 26}{13}) + 2 πn, θ = π + arcsin (\frac{2 26}{13}) + 2 πn

θ = arcsin (- \frac{2 26}{13}) + 2 πn, θ = π + arcsin (\frac{2 26}{13}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{2 26}{13}) + 2 πn, θ = π - arcsin (\frac{2 26}{13}) + 2 πn, θ = arcsin (- \frac{2 26}{13}) + 2 πn, θ = π + arcsin (\frac{2 26}{13}) + 2 πn

Show solutions in decimal form

θ = 0.90183 \dots + 2 πn, θ = π - 0.90183 \dots + 2 πn, θ = - 0.90183 \dots + 2 πn, θ = π + 0.90183 \dots + 2 πn

Graph

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