What is the value of tan(157.5) ?

The value of tan(157.5) is -sqrt(3-2\sqrt{2)}

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Popular >

tan(157.5)

Solution

tan (157. 5^{\circ})

Solution

- 3 - 22

Decimal Notation

- 0.41421 \dots

Solution steps

tan (157. 5^{\circ})

Rewrite using trig identities:

- \frac{1 - cos ( 31 5 ^{\circ} )}{1 + cos ( 31 5 ^{\circ} )}

tan (157. 5^{\circ})

Write

tan (157. 5^{\circ})

tan (\frac{31 5 ^{\circ}}{2})

= tan (\frac{31 5 ^{\circ}}{2})

Use the Half Angle identity:

tan (\frac{θ}{2}) = - \frac{1 - cos ( θ )}{1 + cos ( θ )}

Rewrite using trig identities:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Use the following identity

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

Square both sides

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

Rewrite using trig identities:

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

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

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

Rewrite using trig identities:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

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

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

Simplify

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Substitute

θ

with

\frac{θ}{2}

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

Simplify

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] quadrant I II tan positive negative

tan (\frac{θ}{2}) = - \frac{1 - cos ( θ )}{1 + cos ( θ )}

= - \frac{1 - cos ( 31 5 ^{\circ} )}{1 + cos ( 31 5 ^{\circ} )}

= - \frac{1 - cos ( 31 5 ^{\circ} )}{1 + cos ( 31 5 ^{\circ} )}

Rewrite using trig identities:

cos (31 5^{\circ}) = \frac{2}{2}

cos (31 5^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

cos (31 5^{\circ})

Write

cos (31 5^{\circ})

cos (18 0^{\circ} + 13 5^{\circ})

= cos (18 0^{\circ} + 13 5^{\circ})

Use the Angle Sum identity:

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

= cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

= cos (18 0^{\circ}) cos (13 5^{\circ}) - sin (18 0^{\circ}) sin (13 5^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 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}

= (- 1)

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 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:

sin (18 0^{\circ}) = 0

sin (18 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}

= 0

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 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}

= (- 1) (- \frac{2}{2}) - 0 \cdot \frac{2}{2}

Simplify

= \frac{2}{2}

= - \frac{1 - \frac{2}{2}}{1 + \frac{2}{2}}

Simplify

- \frac{1 - \frac{2}{2}}{1 + \frac{2}{2}} : - 3 - 22

- \frac{1 - \frac{2}{2}}{1 + \frac{2}{2}}

\frac{1 - \frac{2}{2}}{1 + \frac{2}{2}} = \frac{2 - 1}{2 + 1}

\frac{1 - \frac{2}{2}}{1 + \frac{2}{2}}

Join

1 + \frac{2}{2} : \frac{2 + 2}{2}

1 + \frac{2}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 2}{2}

= \frac{1 - \frac{2}{2}}{\frac{2 + 2}{2}}

Join

1 - \frac{2}{2} : \frac{2 - 2}{2}

1 - \frac{2}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

= \frac{\frac{2 - 2}{2}}{\frac{2 + 2}{2}}

Divide fractions:

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

= \frac{( 2 - 2 ) \cdot 2}{2 ( 2 + 2 )}

Cancel the common factor:

2

= \frac{2 - 2}{2 + 2}

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

= \frac{2 ( 2 - 1 )}{2 + 2}

Factor

2 + 2 : 2 (2 + 1)

2 + 2

2 = 22

= 22 + 2

Factor out common term

2

= 2 (2 + 1)

= \frac{2 ( 2 - 1 )}{2 ( 2 + 1 )}

Cancel the common factor:

2

= \frac{2 - 1}{2 + 1}

= - \frac{2 - 1}{1 + 2}

\frac{2 - 1}{2 + 1} = 3 - 22

\frac{2 - 1}{2 + 1}

Multiply by the conjugate

\frac{2 - 1}{2 - 1}

= \frac{( 2 - 1 ) ( 2 - 1 )}{( 2 + 1 ) ( 2 - 1 )}

(2 - 1) (2 - 1) = 3 - 22

(2 - 1) (2 - 1)

Apply exponent rule:

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

(2 - 1) (2 - 1) = (2 - 1)^{1 + 1}

= (2 - 1)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (2 - 1)^{2}

Apply Perfect Square Formula:

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

a = 2, b = 1

= (2)^{2} - 22 \cdot 1 + 1^{2}

Simplify

(2)^{2} - 22 \cdot 1 + 1^{2} : 3 - 22

(2)^{2} - 22 \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 2 \cdot 1 \cdot 2 + 1

(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

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= 2 - 22 + 1

Add the numbers:

2 + 1 = 3

= 3 - 22

= 3 - 22

(2 + 1) (2 - 1) = 1

(2 + 1) (2 - 1)

Apply Difference of Two Squares Formula:

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

a = 2, b = 1

= (2)^{2} - 1^{2}

Simplify

(2)^{2} - 1^{2} : 1

(2)^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(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

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

= \frac{3 - 2 2}{1}

Apply rule

\frac{a}{1} = a

= 3 - 22

= - 3 - 22

= - 3 - 22

Popular Examples

tan((19pi)/6)

tan (\frac{19 π}{6})

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cos (\frac{2 π}{4})

tan(-135)

tan (- 13 5^{\circ})

4cos(pi/6)

4 cos (\frac{π}{6})

csc(-(3pi)/4)

csc (- \frac{3 π}{4})

Frequently Asked Questions (FAQ)

What is the value of tan(157.5) ?
The value of tan(157.5) is -sqrt(3-2\sqrt{2)}