What is the value of csc(510)-4cos(1020)+cos(0) ?

The value of csc(510)-4cos(1020)+cos(0) is 1

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Popular >

csc(510)-4cos(1020)+cos(0)

Solution

csc (51 0^{\circ}) - 4 cos (102 0^{\circ}) + cos (0^{\circ})

Solution

1

Solution steps

csc (51 0^{\circ}) - 4 cos (102 0^{\circ}) + cos (0)

csc (51 0^{\circ}) = csc (15 0^{\circ})

csc (51 0^{\circ})

Rewrite

51 0^{\circ}

36 0^{\circ} + 15 0^{\circ}

= csc (36 0^{\circ} + 15 0^{\circ})

Apply the periodicity of

csc

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

csc (36 0^{\circ} + 15 0^{\circ}) = csc (15 0^{\circ})

= csc (15 0^{\circ})

= csc (15 0^{\circ}) - 4 cos (102 0^{\circ}) + cos (0)

cos (102 0^{\circ}) = cos (30 0^{\circ})

cos (102 0^{\circ})

Rewrite

102 0^{\circ}

36 0^{\circ} \cdot 2 + 30 0^{\circ}

= cos (36 0^{\circ} 2 + 30 0^{\circ})

Apply the periodicity of

cos

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

cos (36 0^{\circ} \cdot 2 + 30 0^{\circ}) = cos (30 0^{\circ})

= cos (30 0^{\circ})

= csc (15 0^{\circ}) - 4 cos (30 0^{\circ}) + cos (0)

Rewrite using trig identities:

csc (15 0^{\circ}) = 2

csc (15 0^{\circ})

Rewrite using trig identities:

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

csc (15 0^{\circ})

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

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

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}

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

Simplify

= 2

Rewrite using trig identities:

cos (30 0^{\circ}) = \frac{1}{2}

cos (30 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\circ})

cos (30 0^{\circ})

Write

cos (30 0^{\circ})

cos (18 0^{\circ} + 12 0^{\circ})

= cos (18 0^{\circ} + 12 0^{\circ})

Use the Angle Sum identity:

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

= cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\circ})

= cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\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 (12 0^{\circ}) = - \frac{1}{2}

cos (12 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{1}{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 (12 0^{\circ}) = \frac{3}{2}

sin (12 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{3}{2}

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

Simplify

= \frac{1}{2}

Use the following trivial identity:

cos (0) = 1

cos (0)

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

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

Simplify

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

2 - 4 \cdot \frac{1}{2} + 1

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

4 \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2 - 2 + 1

Add/Subtract the numbers:

2 - 2 + 1 = 1

= 1

= 1

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Frequently Asked Questions (FAQ)

What is the value of csc(510)-4cos(1020)+cos(0) ?
The value of csc(510)-4cos(1020)+cos(0) is 1