What is the general solution for csc(3θ)=sec(15) ?

The general solution for csc(3θ)=sec(15) is θ=(1.30899…)/3+(360n)/3 ,θ=60-(1.30899…)/3+(360n)/3

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csc(3θ)=sec(15)

Solution

csc (3 θ) = sec (1 5^{\circ})

Solution

θ = \frac{1.30899 \dots}{3} + \frac{36 0 ^{\circ} n}{3}, θ = 6 0^{\circ} - \frac{1.30899 \dots}{3} + \frac{36 0 ^{\circ} n}{3}

Radians

θ = \frac{1.30899 \dots}{3} + \frac{2 π}{3} n, θ = \frac{π}{3} - \frac{1.30899 \dots}{3} + \frac{2 π}{3} n

Solution steps

csc (3 θ) = sec (1 5^{\circ})

sec (1 5^{\circ}) = 6 - 2

sec (1 5^{\circ})

Rewrite using trig identities:

\frac{1}{cos ( 1 5 ^{\circ} )}

sec (1 5^{\circ})

Use the basic trigonometric identity:

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

= \frac{1}{cos ( 1 5 ^{\circ} )}

= \frac{1}{cos ( 1 5 ^{\circ} )}

Rewrite using trig identities:

cos (1 5^{\circ}) = \frac{6 + 2}{4}

cos (1 5^{\circ})

Rewrite using trig identities:

cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

cos (1 5^{\circ})

Write

cos (1 5^{\circ})

cos (4 5^{\circ} - 3 0^{\circ})

= cos (4 5^{\circ} - 3 0^{\circ})

Use the Angle Difference identity:

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

= cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

= cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

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

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

Simplify

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{6 + 2}{4}

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{2 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplify

23 : 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{2 \cdot 1}{2 \cdot 2}

Multiply:

2 \cdot 1 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

= \frac{6}{4} + \frac{2}{4}

Apply rule

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

= \frac{6 + 2}{4}

= \frac{6 + 2}{4}

= \frac{1}{\frac{6 + 2}{4}}

Simplify

\frac{1}{\frac{6 + 2}{4}} : 6 - 2

\frac{1}{\frac{6 + 2}{4}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{4}{6 + 2}

Rationalize

\frac{4}{6 + 2} : 6 - 2

\frac{4}{6 + 2}

Multiply by the conjugate

\frac{6 - 2}{6 - 2}

= \frac{4 ( 6 - 2 )}{( 6 + 2 ) ( 6 - 2 )}

(6 + 2) (6 - 2) = 4

(6 + 2) (6 - 2)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 6, b = 2

= (6)^{2} - (2)^{2}

Simplify

(6)^{2} - (2)^{2} : 4

(6)^{2} - (2)^{2}

(6)^{2} = 6

(6)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 6

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 6 - 2

Subtract the numbers:

6 - 2 = 4

= 4

= 4

= \frac{4 ( 6 - 2 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 6 - 2

= 6 - 2

= 6 - 2

csc (3 θ) = 6 - 2

Apply trig inverse properties

csc (3 θ) = 6 - 2

General solutions for

csc (3 θ) = 6 - 2

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

3 θ = arccsc (6 - 2) + 36 0^{\circ} n, 3 θ = 18 0^{\circ} - arccsc (6 - 2) + 36 0^{\circ} n

3 θ = arccsc (6 - 2) + 36 0^{\circ} n, 3 θ = 18 0^{\circ} - arccsc (6 - 2) + 36 0^{\circ} n

Solve

3 θ = arccsc (6 - 2) + 36 0^{\circ} n : θ = \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

3 θ = arccsc (6 - 2) + 36 0^{\circ} n

Divide both sides by

3

3 θ = arccsc (6 - 2) + 36 0^{\circ} n

Divide both sides by

3

\frac{3 θ}{3} = \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

Simplify

θ = \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

θ = \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

Solve

3 θ = 18 0^{\circ} - arccsc (6 - 2) + 36 0^{\circ} n : θ = 6 0^{\circ} - \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

3 θ = 18 0^{\circ} - arccsc (6 - 2) + 36 0^{\circ} n

Divide both sides by

3

3 θ = 18 0^{\circ} - arccsc (6 - 2) + 36 0^{\circ} n

Divide both sides by

3

\frac{3 θ}{3} = 6 0^{\circ} - \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

Simplify

θ = 6 0^{\circ} - \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

θ = 6 0^{\circ} - \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

θ = \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}, θ = 6 0^{\circ} - \frac{arccsc ( 6 - 2 )}{3} + \frac{36 0 ^{\circ} n}{3}

Show solutions in decimal form

θ = \frac{1.30899 \dots}{3} + \frac{36 0 ^{\circ} n}{3}, θ = 6 0^{\circ} - \frac{1.30899 \dots}{3} + \frac{36 0 ^{\circ} n}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for csc(3θ)=sec(15) ?
The general solution for csc(3θ)=sec(15) is θ=(1.30899…)/3+(360n)/3 ,θ=60-(1.30899…)/3+(360n)/3