What is the value of sec(600)+cot(315) ?

The value of sec(600)+cot(315) is -3

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sec(600)+cot(315)

Solution

sec (60 0^{\circ}) + cot (31 5^{\circ})

Solution

- 3

Solution steps

sec (60 0^{\circ}) + cot (31 5^{\circ})

sec (60 0^{\circ}) = sec (24 0^{\circ})

sec (60 0^{\circ})

Rewrite

60 0^{\circ}

36 0^{\circ} + 24 0^{\circ}

= sec (36 0^{\circ} + 24 0^{\circ})

Apply the periodicity of

sec

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

sec (36 0^{\circ} + 24 0^{\circ}) = sec (24 0^{\circ})

= sec (24 0^{\circ})

= sec (24 0^{\circ}) + cot (31 5^{\circ})

cot (31 5^{\circ}) = cot (13 5^{\circ})

cot (31 5^{\circ})

Rewrite

31 5^{\circ}

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

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

Apply the periodicity of

cot

cot (x + 18 0^{\circ}) = cot (x)

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

= cot (13 5^{\circ})

= sec (24 0^{\circ}) + cot (13 5^{\circ})

Rewrite using trig identities:

sec (24 0^{\circ}) = - 2

sec (24 0^{\circ})

Rewrite using trig identities:

\frac{1}{cos ( 24 0 ^{\circ} )}

sec (24 0^{\circ})

Use the basic trigonometric identity:

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

= \frac{1}{cos ( 24 0 ^{\circ} )}

= \frac{1}{cos ( 24 0 ^{\circ} )}

Rewrite using trig identities:

cos (24 0^{\circ}) = - \frac{1}{2}

cos (24 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

cos (24 0^{\circ})

Write

cos (24 0^{\circ})

cos (18 0^{\circ} + 6 0^{\circ})

= cos (18 0^{\circ} + 6 0^{\circ})

Use the Angle Sum identity:

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

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 0^{\circ})

= cos (18 0^{\circ}) cos (6 0^{\circ}) - sin (18 0^{\circ}) sin (6 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 (6 0^{\circ}) = \frac{1}{2}

cos (6 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 (6 0^{\circ}) = \frac{3}{2}

sin (6 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}

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

Simplify

= - 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

= - 2 - 1

Simplify

= - 3

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Frequently Asked Questions (FAQ)

What is the value of sec(600)+cot(315) ?
The value of sec(600)+cot(315) is -3