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Popular Trigonometry >

10tan(θ)-4.9((10)/(13cos(θ)))^2=0

Solution

10 tan (θ) - 4.9 (\frac{10}{13 cos ( θ )})^{2} = 0

Solution

θ = \frac{0.61858 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.61858 \dots}{2} + πn

Degrees

θ = 17.72110 \dots^{\circ} + 18 0^{\circ} n, θ = 72.27889 \dots^{\circ} + 18 0^{\circ} n

Solution steps

10 tan (θ) - 4.9 (\frac{10}{13 cos ( θ )})^{2} = 0

Express with sin, cos

- (\frac{10}{13 cos ( θ )})^{2} \cdot 4.9 + 10 tan (θ)

Use the basic trigonometric identity:

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

= - (\frac{10}{13 cos ( θ )})^{2} \cdot 4.9 + 10 \cdot \frac{sin ( θ )}{cos ( θ )}

Simplify

- (\frac{10}{13 cos ( θ )})^{2} \cdot 4.9 + 10 \cdot \frac{sin ( θ )}{cos ( θ )} : \frac{- 2.89940 \dots + 10 sin ( θ ) cos ( θ )}{cos ^{2} ( θ )}

- (\frac{10}{13 cos ( θ )})^{2} \cdot 4.9 + 10 \cdot \frac{sin ( θ )}{cos ( θ )}

(\frac{10}{13 cos ( θ )})^{2} \cdot 4.9 = \frac{2.89940 \dots}{cos ^{2} ( θ )}

(\frac{10}{13 cos ( θ )})^{2} \cdot 4.9

(\frac{10}{13 cos ( θ )})^{2} = \frac{1 0 ^{2}}{1 3 ^{2} cos ^{2} ( θ )}

(\frac{10}{13 cos ( θ )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 0 ^{2}}{( 13 cos ( θ ) ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(13 cos (θ))^{2} = 1 3^{2} cos^{2} (θ)

= \frac{1 0 ^{2}}{1 3 ^{2} cos ^{2} ( θ )}

= 4.9 \cdot \frac{1 0 ^{2}}{1 3 ^{2} cos ^{2} ( θ )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 0 ^{2} \cdot 4.9}{1 3 ^{2} cos ^{2} ( θ )}

1 0^{2} \cdot 4.9 = 490

1 0^{2} \cdot 4.9

1 0^{2} = 100

= 100 \cdot 4.9

Multiply the numbers:

100 \cdot 4.9 = 490

= 490

= \frac{490}{1 3 ^{2} cos ^{2} ( θ )}

1 3^{2} = 169

= \frac{490}{169 cos ^{2} ( θ )}

Convert element to a decimal form

\frac{490}{169} = 2.89940 \dots

= \frac{2.89940 \dots}{cos ^{2} ( θ )}

10 \cdot \frac{sin ( θ )}{cos ( θ )} = \frac{10 sin ( θ )}{cos ( θ )}

10 \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( θ ) \cdot 10}{cos ( θ )}

= - \frac{2.89940 \dots}{cos ^{2} ( θ )} + \frac{10 sin ( θ )}{cos ( θ )}

Least Common Multiplier of

cos^{2} (θ), cos (θ) : cos^{2} (θ)

cos^{2} (θ), cos (θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos^{2} (θ)

cos (θ)

= cos^{2} (θ)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos^{2} (θ)

For

\frac{sin ( θ ) \cdot 10}{cos ( θ )} :

multiply the denominator and numerator by

cos (θ)

\frac{sin ( θ ) \cdot 10}{cos ( θ )} = \frac{sin ( θ ) \cdot 10 cos ( θ )}{cos ( θ ) cos ( θ )} = \frac{sin ( θ ) \cdot 10 cos ( θ )}{cos ^{2} ( θ )}

= - \frac{2.89940 \dots}{cos ^{2} ( θ )} + \frac{sin ( θ ) \cdot 10 cos ( θ )}{cos ^{2} ( θ )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2.89940 \dots + sin ( θ ) \cdot 10 cos ( θ )}{cos ^{2} ( θ )}

= \frac{- 2.89940 \dots + 10 sin ( θ ) cos ( θ )}{cos ^{2} ( θ )}

\frac{- 2.89940 \dots + 10 cos ( θ ) sin ( θ )}{cos ^{2} ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2.89940 \dots + 10 cos (θ) sin (θ) = 0

Rewrite using trig identities

- 2.89940 \dots + 10 cos (θ) sin (θ)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 2.89940 \dots + 10 \cdot \frac{sin ( 2 θ )}{2}

- 2.89940 \dots + 10 \cdot \frac{sin ( 2 θ )}{2} = 0

10 \cdot \frac{sin ( 2 θ )}{2} = 5 sin (2 θ)

10 \cdot \frac{sin ( 2 θ )}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 θ ) \cdot 10}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 sin (2 θ)

- 2.89940 \dots + 5 sin (2 θ) = 0

Move

2.89940 \dots

to the right side

- 2.89940 \dots + 5 sin (2 θ) = 0

Add

2.89940 \dots

to both sides

- 2.89940 \dots + 5 sin (2 θ) + 2.89940 \dots = 0 + 2.89940 \dots

Simplify

5 sin (2 θ) = 2.89940 \dots

5 sin (2 θ) = 2.89940 \dots

Divide both sides by

5

5 sin (2 θ) = 2.89940 \dots

Divide both sides by

5

\frac{5 sin ( 2 θ )}{5} = \frac{2.89940 \dots}{5}

Simplify

sin (2 θ) = 0.57988 \dots

sin (2 θ) = 0.57988 \dots

Apply trig inverse properties

sin (2 θ) = 0.57988 \dots

General solutions for

sin (2 θ) = 0.57988 \dots

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

2 θ = arcsin (0.57988 \dots) + 2 πn, 2 θ = π - arcsin (0.57988 \dots) + 2 πn

2 θ = arcsin (0.57988 \dots) + 2 πn, 2 θ = π - arcsin (0.57988 \dots) + 2 πn

Solve

2 θ = arcsin (0.57988 \dots) + 2 πn : θ = \frac{arcsin ( 0.57988 \dots )}{2} + πn

2 θ = arcsin (0.57988 \dots) + 2 πn

Divide both sides by

2

2 θ = arcsin (0.57988 \dots) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{arcsin ( 0.57988 \dots )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{arcsin ( 0.57988 \dots )}{2} + πn

θ = \frac{arcsin ( 0.57988 \dots )}{2} + πn

Solve

2 θ = π - arcsin (0.57988 \dots) + 2 πn : θ = \frac{π}{2} - \frac{arcsin ( 0.57988 \dots )}{2} + πn

2 θ = π - arcsin (0.57988 \dots) + 2 πn

Divide both sides by

2

2 θ = π - arcsin (0.57988 \dots) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} - \frac{arcsin ( 0.57988 \dots )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} - \frac{arcsin ( 0.57988 \dots )}{2} + πn

θ = \frac{π}{2} - \frac{arcsin ( 0.57988 \dots )}{2} + πn

θ = \frac{arcsin ( 0.57988 \dots )}{2} + πn, θ = \frac{π}{2} - \frac{arcsin ( 0.57988 \dots )}{2} + πn

Show solutions in decimal form

θ = \frac{0.61858 \dots}{2} + πn, θ = \frac{π}{2} - \frac{0.61858 \dots}{2} + πn

Graph

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