What is the general solution for sin(x)+cos(x)=sec(x) ?

The general solution for sin(x)+cos(x)=sec(x) is x=2pin,x=pi+2pin,x= pi/4+pin

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sin(x)+cos(x)=sec(x)

Solution

sin (x) + cos (x) = sec (x)

Solution

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) + cos (x) = sec (x)

Subtract

sec (x)

from both sides

sin (x) + cos (x) - sec (x) = 0

Express with sin, cos

cos (x) - sec (x) + sin (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= cos (x) - \frac{1}{cos ( x )} + sin (x)

Simplify

cos (x) - \frac{1}{cos ( x )} + sin (x) : \frac{cos ^{2} ( x ) - 1 + sin ( x ) cos ( x )}{cos ( x )}

cos (x) - \frac{1}{cos ( x )} + sin (x)

Convert element to fraction:

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

= \frac{cos ( x ) cos ( x )}{cos ( x )} - \frac{1}{cos ( x )} + \frac{sin ( x ) cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( x ) cos ( x ) - 1 + sin ( x ) cos ( x )}{cos ( x )}

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

cos (x) cos (x) - 1 + sin (x) cos (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

\frac{- 1 + cos ^{2} ( x ) + cos ( x ) sin ( x )}{cos ( x )} = 0

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

- 1 + cos^{2} (x) + cos (x) sin (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

- sin^{2} (x) + cos (x) sin (x) = 0

Factor

- sin^{2} (x) + cos (x) sin (x) : sin (x) (- sin (x) + cos (x))

- sin^{2} (x) + cos (x) sin (x)

Apply exponent rule:

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

sin^{2} (x) = sin (x) sin (x)

= - sin (x) sin (x) + sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (- sin (x) + cos (x))

sin (x) (- sin (x) + cos (x)) = 0

Solving each part separately

sin (x) = 0 or - sin (x) + cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- sin (x) + cos (x) = 0 : x = \frac{π}{4} + πn

- sin (x) + cos (x) = 0

Rewrite using trig identities

- sin (x) + cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{- sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- \frac{sin ( x )}{cos ( x )} + 1 = 0

Use the basic trigonometric identity:

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

- tan (x) + 1 = 0

- tan (x) + 1 = 0

Move

1

to the right side

- tan (x) + 1 = 0

Subtract

1

from both sides

- tan (x) + 1 - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+cos(x)=sec(x) ?
The general solution for sin(x)+cos(x)=sec(x) is x=2pin,x=pi+2pin,x= pi/4+pin