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Popular Trigonometry >

110^6=(400cos(θ)+700)^2+(400*sin(θ))^2

Solution

1 \cdot 1 0^{6} = (400 \cdot cos (θ) + 700)^{2} + (400 \cdot sin (θ))^{2}

Solution

θ = 0.89566 \dots + 2 πn, θ = 2 π - 0.89566 \dots + 2 πn

Degrees

θ = 51.31781 \dots^{\circ} + 36 0^{\circ} n, θ = 308.68218 \dots^{\circ} + 36 0^{\circ} n

Solution steps

1 \cdot 1 0^{6} = (400 cos (θ) + 700)^{2} + (400 sin (θ))^{2}

Switch sides

(400 cos (θ) + 700)^{2} + (400 sin (θ))^{2} = 1 \cdot 1 0^{6}

Subtract

11 0^{6}

from both sides

160000 cos^{2} (θ) + 160000 sin^{2} (θ) + 560000 cos (θ) - 510000 = 0

Rewrite using trig identities

- 510000 + 160000 cos^{2} (θ) + 160000 sin^{2} (θ) + 560000 cos (θ)

Use the Pythagorean identity:

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

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

= - 510000 + 160000 (1 - sin^{2} (θ)) + 160000 sin^{2} (θ) + 560000 cos (θ)

Simplify

- 510000 + 160000 (1 - sin^{2} (θ)) + 160000 sin^{2} (θ) + 560000 cos (θ) : 560000 cos (θ) - 350000

- 510000 + 160000 (1 - sin^{2} (θ)) + 160000 sin^{2} (θ) + 560000 cos (θ)

Expand

160000 (1 - sin^{2} (θ)) : 160000 - 160000 sin^{2} (θ)

160000 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 160000, b = 1, c = sin^{2} (θ)

= 160000 \cdot 1 - 160000 sin^{2} (θ)

Multiply the numbers:

160000 \cdot 1 = 160000

= 160000 - 160000 sin^{2} (θ)

= - 510000 + 160000 - 160000 sin^{2} (θ) + 160000 sin^{2} (θ) + 560000 cos (θ)

Simplify

- 510000 + 160000 - 160000 sin^{2} (θ) + 160000 sin^{2} (θ) + 560000 cos (θ) : 560000 cos (θ) - 350000

- 510000 + 160000 - 160000 sin^{2} (θ) + 160000 sin^{2} (θ) + 560000 cos (θ)

Add similar elements:

- 160000 sin^{2} (θ) + 160000 sin^{2} (θ) = 0

= - 510000 + 160000 + 560000 cos (θ)

Add/Subtract the numbers:

- 510000 + 160000 = - 350000

= 560000 cos (θ) - 350000

= 560000 cos (θ) - 350000

= 560000 cos (θ) - 350000

- 350000 + 560000 cos (θ) = 0

Move

350000

to the right side

- 350000 + 560000 cos (θ) = 0

Add

350000

to both sides

- 350000 + 560000 cos (θ) + 350000 = 0 + 350000

Simplify

560000 cos (θ) = 350000

560000 cos (θ) = 350000

Divide both sides by

560000

560000 cos (θ) = 350000

Divide both sides by

560000

\frac{560000 cos ( θ )}{560000} = \frac{350000}{560000}

Simplify

cos (θ) = \frac{5}{8}

cos (θ) = \frac{5}{8}

Apply trig inverse properties

cos (θ) = \frac{5}{8}

General solutions for

cos (θ) = \frac{5}{8}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{5}{8}) + 2 πn, θ = 2 π - arccos (\frac{5}{8}) + 2 πn

θ = arccos (\frac{5}{8}) + 2 πn, θ = 2 π - arccos (\frac{5}{8}) + 2 πn

Show solutions in decimal form

θ = 0.89566 \dots + 2 πn, θ = 2 π - 0.89566 \dots + 2 πn

Graph

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1*10^6=(400*cos(θ)+700)^2+(400*sin(θ))^2

Solution

Solution

Graph

Popular Examples

110^6=(400cos(θ)+700)^2+(400*sin(θ))^2