What is the value of tan(1215)+sin^2(300) ?

The value of tan(1215)+sin^2(300) is -1/4

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tan(1215)+sin^2(300)

Solution

tan (121 5^{\circ}) + sin^{2} (30 0^{\circ})

Solution

- \frac{1}{4}

Decimal Notation

- 0.25

Solution steps

tan (121 5^{\circ}) + sin^{2} (30 0^{\circ})

Rewrite using trig identities:

sin (30 0^{\circ}) = 2 sin (15 0^{\circ}) cos (15 0^{\circ})

sin (30 0^{\circ})

Write

sin (30 0^{\circ})

sin (2 \cdot 15 0^{\circ})

= sin (2 \cdot 15 0^{\circ})

Use the Double Angle identity:

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

= 2 sin (15 0^{\circ}) cos (15 0^{\circ})

= tan (121 5^{\circ}) + (2 sin (15 0^{\circ}) cos (15 0^{\circ}))^{2}

Simplify

= tan (121 5^{\circ}) + 4 sin^{2} (15 0^{\circ}) cos^{2} (15 0^{\circ})

tan (121 5^{\circ}) = tan (13 5^{\circ})

tan (121 5^{\circ})

Rewrite

121 5^{\circ}

18 0^{\circ} \cdot 6 + 13 5^{\circ}

= tan (18 0^{\circ} 6 + 13 5^{\circ})

Apply the periodicity of

tan

tan (x + 18 0^{\circ} \cdot k) = tan (x)

tan (18 0^{\circ} \cdot 6 + 13 5^{\circ}) = tan (13 5^{\circ})

= tan (13 5^{\circ})

= tan (13 5^{\circ}) + 4 sin^{2} (15 0^{\circ}) cos^{2} (15 0^{\circ})

Rewrite using trig identities:

tan (13 5^{\circ}) = - 1

tan (13 5^{\circ})

Rewrite using trig identities:

\frac{sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

tan (13 5^{\circ})

Use the basic trigonometric identity:

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

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

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

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}

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}

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

Simplify

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

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

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

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

Divide fractions:

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

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

Cancel the common factor:

2

= - \frac{2}{2}

Cancel the common factor:

2

= - 1

= - 1

Use the following trivial identity:

sin (15 0^{\circ}) = \frac{1}{2}

sin (15 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}

Use the following trivial identity:

cos (15 0^{\circ}) = - \frac{3}{2}

cos (15 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}

= - 1 + 4 (\frac{1}{2})^{2} (- \frac{3}{2})^{2}

Simplify

- 1 + 4 (\frac{1}{2})^{2} (- \frac{3}{2})^{2} : - \frac{1}{4}

- 1 + 4 (\frac{1}{2})^{2} (- \frac{3}{2})^{2}

4 (\frac{1}{2})^{2} (- \frac{3}{2})^{2} = \frac{3}{4}

4 (\frac{1}{2})^{2} (- \frac{3}{2})^{2}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= 4 \cdot \frac{1}{2 ^{2}} (- \frac{3}{2})^{2}

(- \frac{3}{2})^{2} = \frac{3}{2 ^{2}}

(- \frac{3}{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{3}{2})^{2} = (\frac{3}{2})^{2}

= (\frac{3}{2})^{2}

Apply exponent rule:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= \frac{3}{2 ^{2}}

= 4 \cdot \frac{1}{2 ^{2}} \cdot \frac{3}{2 ^{2}}

Multiply fractions:

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

= \frac{1 \cdot 3 \cdot 4}{2 ^{2} \cdot 2 ^{2}}

Multiply the numbers:

1 \cdot 3 \cdot 4 = 12

= \frac{12}{2 ^{2} \cdot 2 ^{2}}

2^{2} \cdot 2^{2} = 2^{4}

2^{2} \cdot 2^{2}

Apply exponent rule:

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

2^{2} \cdot 2^{2} = 2^{2 + 2}

= 2^{2 + 2}

Add the numbers:

2 + 2 = 4

= 2^{4}

= \frac{12}{2 ^{4}}

Factor

12 : 2^{2} \cdot 3

Factor

12 = 2^{2} \cdot 3

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

Cancel

\frac{2 ^{2} \cdot 3}{2 ^{4}} : \frac{3}{2 ^{2}}

\frac{2 ^{2} \cdot 3}{2 ^{4}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{2}}{2 ^{4}} = \frac{1}{2 ^{4 - 2}}

= \frac{3}{2 ^{4 - 2}}

Subtract the numbers:

4 - 2 = 2

= \frac{3}{2 ^{2}}

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= - 1 + \frac{3}{4}

Convert element to fraction:

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

= - \frac{1 \cdot 4}{4} + \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 4 + 3}{4}

- 1 \cdot 4 + 3 = - 1

- 1 \cdot 4 + 3

Multiply the numbers:

1 \cdot 4 = 4

= - 4 + 3

Add/Subtract the numbers:

- 4 + 3 = - 1

= - 1

= \frac{- 1}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{4}

= - \frac{1}{4}

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Frequently Asked Questions (FAQ)

What is the value of tan(1215)+sin^2(300) ?
The value of tan(1215)+sin^2(300) is -1/4