What is the general solution for -0.3cos(θ)=cos(2.2+θ) ?

The general solution for -0.3cos(θ)=cos(2.2+θ) is θ=-0.34275…+pin

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Popular Trigonometry >

-0.3cos(θ)=cos(2.2+θ)

Solution

- 0.3 cos (θ) = cos (2.2 + θ)

Solution

θ = - 0.34275 \dots + πn

Degrees

θ = - 19.63826 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 0.3 cos (θ) = cos (2.2 + θ)

Rewrite using trig identities

- 0.3 cos (θ) = cos (2.2 + θ)

Rewrite using trig identities

cos (2.2 + θ)

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2.2) cos (θ) - sin (2.2) sin (θ)

Simplify

cos (2.2) cos (θ) - sin (2.2) sin (θ) : - 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

cos (2.2) cos (θ) - sin (2.2) sin (θ)

Simplify

cos (2.2) : - 0.58850 \dots

cos (2.2)

cos (2.2) = - 0.58850 \dots

= - 0.58850 \dots

= - 0.58850 \dots cos (θ) - sin (2.2) sin (θ)

Simplify

sin (2.2) : 0.80849 \dots

sin (2.2)

sin (2.2) = 0.80849 \dots

= 0.80849 \dots

= - 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

= - 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

- 0.3 cos (θ) = - 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

- 0.3 cos (θ) = - 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

Subtract

- 0.58850 \dots cos (θ) - 0.80849 \dots sin (θ)

from both sides

0.28850 \dots cos (θ) + 0.80849 \dots sin (θ) = 0

Rewrite using trig identities

0.28850 \dots cos (θ) + 0.80849 \dots sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{0.28850 \dots cos ( θ ) + 0.80849 \dots sin ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

0.28850 \dots + \frac{0.80849 \dots sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

0.28850 \dots + 0.80849 \dots tan (θ) = 0

0.28850 \dots + 0.80849 \dots tan (θ) = 0

Move

0.28850 \dots

to the right side

0.28850 \dots + 0.80849 \dots tan (θ) = 0

Subtract

0.28850 \dots

from both sides

0.28850 \dots + 0.80849 \dots tan (θ) - 0.28850 \dots = 0 - 0.28850 \dots

Simplify

0.80849 \dots tan (θ) = - 0.28850 \dots

0.80849 \dots tan (θ) = - 0.28850 \dots

Divide both sides by

0.80849 \dots

0.80849 \dots tan (θ) = - 0.28850 \dots

Divide both sides by

0.80849 \dots

\frac{0.80849 \dots tan ( θ )}{0.80849 \dots} = \frac{- 0.28850 \dots}{0.80849 \dots}

Simplify

\frac{0.80849 \dots tan ( θ )}{0.80849 \dots} = \frac{- 0.28850 \dots}{0.80849 \dots}

Simplify

\frac{0.80849 \dots tan ( θ )}{0.80849 \dots} : tan (θ)

\frac{0.80849 \dots tan ( θ )}{0.80849 \dots}

Cancel the common factor:

0.80849 \dots

= tan (θ)

Simplify

\frac{- 0.28850 \dots}{0.80849 \dots} : - 0.35683 \dots

\frac{- 0.28850 \dots}{0.80849 \dots}

Apply the fraction rule:

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

= - \frac{0.28850 \dots}{0.80849 \dots}

Divide the numbers:

\frac{0.28850 \dots}{0.80849 \dots} = 0.35683 \dots

= - 0.35683 \dots

tan (θ) = - 0.35683 \dots

tan (θ) = - 0.35683 \dots

tan (θ) = - 0.35683 \dots

Apply trig inverse properties

tan (θ) = - 0.35683 \dots

General solutions for

tan (θ) = - 0.35683 \dots

tan (x) = - a \Rightarrow x = arctan (- a) + πn

θ = arctan (- 0.35683 \dots) + πn

θ = arctan (- 0.35683 \dots) + πn

Show solutions in decimal form

θ = - 0.34275 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -0.3cos(θ)=cos(2.2+θ) ?
The general solution for -0.3cos(θ)=cos(2.2+θ) is θ=-0.34275…+pin