What is the value of cot^2(135)-sin(210)+5cos^2(225) ?

The value of cot^2(135)-sin(210)+5cos^2(225) is 4

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cot^2(135)-sin(210)+5cos^2(225)

Solution

cot^{2} (13 5^{\circ}) - sin (21 0^{\circ}) + 5 cos^{2} (22 5^{\circ})

Solution

4

Solution steps

cot^{2} (13 5^{\circ}) - sin (21 0^{\circ}) + 5 cos^{2} (22 5^{\circ})

Rewrite using trig identities:

cos^{2} (22 5^{\circ}) = \frac{1 + cos ( 9 0 ^{\circ} )}{2}

cos^{2} (22 5^{\circ})

Use the following identity:

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}

= \frac{1 + cos ( 2 \cdot 22 5 ^{\circ} )}{2}

Simplify

= \frac{1 + cos ( 45 0 ^{\circ} )}{2}

cos (45 0^{\circ}) = cos (9 0^{\circ})

cos (45 0^{\circ})

Rewrite

45 0^{\circ}

36 0^{\circ} + 9 0^{\circ}

= cos (36 0^{\circ} + 9 0^{\circ})

Apply the periodicity of

cos

cos (x + 36 0^{\circ}) = cos (x)

cos (36 0^{\circ} + 9 0^{\circ}) = cos (9 0^{\circ})

= cos (9 0^{\circ})

= \frac{1 + cos ( 9 0 ^{\circ} )}{2}

= cot^{2} (13 5^{\circ}) - sin (21 0^{\circ}) + 5 \cdot \frac{1 + cos ( 9 0 ^{\circ} )}{2}

Simplify

= \frac{2 cot ^{2} ( 13 5 ^{\circ} ) - 2 sin ( 21 0 ^{\circ} ) + 5 ( 1 + cos ( 9 0 ^{\circ} ) )}{2}

Use the following trivial identity:

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

cot (13 5^{\circ})

cot (x)

periodicity table with

18 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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= - 1

Rewrite using trig identities:

sin (21 0^{\circ}) = - \frac{1}{2}

sin (21 0^{\circ})

Rewrite using trig identities:

sin (18 0^{\circ}) cos (3 0^{\circ}) + cos (18 0^{\circ}) sin (3 0^{\circ})

sin (21 0^{\circ})

Write

sin (21 0^{\circ})

sin (18 0^{\circ} + 3 0^{\circ})

= sin (18 0^{\circ} + 3 0^{\circ})

Use the Angle Sum identity:

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

= sin (18 0^{\circ}) cos (3 0^{\circ}) + cos (18 0^{\circ}) sin (3 0^{\circ})

= sin (18 0^{\circ}) cos (3 0^{\circ}) + cos (18 0^{\circ}) sin (3 0^{\circ})

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:

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:

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:

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}

= 0 \cdot \frac{3}{2} + (- 1) \frac{1}{2}

Simplify

= - \frac{1}{2}

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (9 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}

= 0

= \frac{2 ( - 1 ) ^{2} - 2 ( - \frac{1}{2} ) + 5 ( 1 + 0 )}{2}

Simplify

\frac{2 ( - 1 ) ^{2} - 2 ( - \frac{1}{2} ) + 5 ( 1 + 0 )}{2} : 4

\frac{2 ( - 1 ) ^{2} - 2 ( - \frac{1}{2} ) + 5 ( 1 + 0 )}{2}

Apply rule

- (- a) = a

= \frac{2 ( - 1 ) ^{2} + 2 \cdot \frac{1}{2} + 5 ( 1 + 0 )}{2}

2 (- 1)^{2} + 2 \cdot \frac{1}{2} + 5 (1 + 0) = 2 \cdot \frac{1}{2} + 7

2 (- 1)^{2} + 2 \cdot \frac{1}{2} + 5 (1 + 0)

2 (- 1)^{2} = 2

2 (- 1)^{2}

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

= 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2

5 (1 + 0) = 5

5 (1 + 0)

Add the numbers:

1 + 0 = 1

= 5 \cdot 1

Multiply the numbers:

5 \cdot 1 = 5

= 5

= 2 + 2 \cdot \frac{1}{2} + 5

Add the numbers:

2 + 5 = 7

= 2 \cdot \frac{1}{2} + 7

= \frac{2 \cdot \frac{1}{2} + 7}{2}

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{1 + 7}{2}

Add the numbers:

1 + 7 = 8

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

= 4

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Frequently Asked Questions (FAQ)

What is the value of cot^2(135)-sin(210)+5cos^2(225) ?
The value of cot^2(135)-sin(210)+5cos^2(225) is 4