What is the value of sec(135)tan(210)sin(240) ?

The value of sec(135)tan(210)sin(240) is (sqrt(2))/2

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sec(135)tan(210)sin(240)

Solution

sec (13 5^{\circ}) tan (21 0^{\circ}) sin (24 0^{\circ})

Solution

\frac{2}{2}

Decimal Notation

0.70710 \dots

Solution steps

sec (13 5^{\circ}) tan (21 0^{\circ}) sin (24 0^{\circ})

tan (21 0^{\circ}) = tan (3 0^{\circ})

tan (21 0^{\circ})

Rewrite

21 0^{\circ}

18 0^{\circ} + 3 0^{\circ}

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

Apply the periodicity of

tan

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

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

= tan (3 0^{\circ})

= sec (13 5^{\circ}) tan (3 0^{\circ}) sin (24 0^{\circ})

Rewrite using trig identities:

sin (24 0^{\circ}) = 2 sin (12 0^{\circ}) cos (12 0^{\circ})

sin (24 0^{\circ})

Write

sin (24 0^{\circ})

sin (2 \cdot 12 0^{\circ})

= sin (2 \cdot 12 0^{\circ})

Use the Double Angle identity:

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

= 2 sin (12 0^{\circ}) cos (12 0^{\circ})

= sec (13 5^{\circ}) tan (3 0^{\circ}) \cdot 2 sin (12 0^{\circ}) cos (12 0^{\circ})

Use the following trivial identity:

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

sec (13 5^{\circ})

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= - 2

Use the following trivial identity:

tan (3 0^{\circ}) = \frac{3}{3}

tan (3 0^{\circ})

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

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}

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}

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

Simplify

(- 2) \frac{3}{3} \cdot 2 \cdot \frac{3}{2} (- \frac{1}{2}) : \frac{2}{2}

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

33 \cdot 1 \cdot 2 = 32

33 \cdot 1 \cdot 2

Apply radical rule:

a a = a

33 = 3

= 3 \cdot 1 \cdot 2

Multiply the numbers:

3 \cdot 1 = 3

= 32

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

Multiply the numbers:

3 \cdot 2 = 6

= \frac{3 2}{6}

Cancel the common factor:

3

= \frac{2}{2}

= \frac{2}{2}

Popular Examples

arcsin(-0.331)

arcsin (- 0.331)

-2sin(-pi/7)

- 2 sin (- \frac{π}{7})

1sin(15)

1 sin (1 5^{\circ})

1/(sin(150))

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

2*sin(120)

2 \cdot sin (12 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of sec(135)tan(210)sin(240) ?
The value of sec(135)tan(210)sin(240) is (sqrt(2))/2