What is the general solution for tan(x)= 3/(10cos(x)),0<x<90 ?

The general solution for tan(x)= 3/(10cos(x)),0<x<90 is x=0.30469…

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Popular Trigonometry >

tan(x)= 3/(10cos(x)),0<x<90

Solution

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

Solution

x = 0.30469 \dots

Solution steps

tan (x) = \frac{3}{10 cos ( x )}, 0^{\circ} < x < 9 0^{\circ}

Subtract

\frac{3}{10 cos ( x )}

from both sides

tan (x) - \frac{3}{10 cos ( x )} = 0

Simplify

tan (x) - \frac{3}{10 cos ( x )} : \frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )}

tan (x) - \frac{3}{10 cos ( x )}

Convert element to fraction:

tan (x) = \frac{tan ( x ) 10 cos ( x )}{10 cos ( x )}

= \frac{tan ( x ) \cdot 10 cos ( x )}{10 cos ( x )} - \frac{3}{10 cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( x ) \cdot 10 cos ( x ) - 3}{10 cos ( x )}

\frac{10 tan ( x ) cos ( x ) - 3}{10 cos ( x )} = 0

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

10 tan (x) cos (x) - 3 = 0

Rewrite using trig identities

10 tan (x) cos (x) - 3

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

cos (x) \frac{sin ( x )}{cos ( x )} : sin (x)

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x)

= - 3 + 10 sin (x)

- 3 + 10 sin (x) = 0

Move

3

to the right side

- 3 + 10 sin (x) = 0

Add

3

to both sides

- 3 + 10 sin (x) + 3 = 0 + 3

Simplify

10 sin (x) = 3

10 sin (x) = 3

Divide both sides by

10

10 sin (x) = 3

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{3}{10}

Simplify

sin (x) = \frac{3}{10}

sin (x) = \frac{3}{10}

Apply trig inverse properties

sin (x) = \frac{3}{10}

General solutions for

sin (x) = \frac{3}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

x = arcsin (\frac{3}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{10}) + 36 0^{\circ} n

Solutions for the range

0 < x < 9 0^{\circ}

x = arcsin (\frac{3}{10})

Show solutions in decimal form

x = 0.30469 \dots

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)= 3/(10cos(x)),0<x<90 ?
The general solution for tan(x)= 3/(10cos(x)),0<x<90 is x=0.30469…