What is the value of 3sec^2((7pi)/6) ?

The value of 3sec^2((7pi)/6) is 4

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3sec^2((7pi)/6)

3 sec^{2} (\frac{7 π}{6})

4

Solution steps

3 sec^{2} (\frac{7 π}{6})

Rewrite using trig identities:

sec^{2} (\frac{7 π}{6}) = tan^{2} (\frac{π}{6}) + 1

sec^{2} (\frac{7 π}{6})

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

= tan^{2} (\frac{7 π}{6}) + 1

tan (\frac{7 π}{6}) = tan (\frac{π}{6})

tan (\frac{7 π}{6})

Rewrite

\frac{7 π}{6}

π + \frac{π}{6}

= tan (π + \frac{π}{6})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{π}{6}) = tan (\frac{π}{6})

= tan (\frac{π}{6})

= tan^{2} (\frac{π}{6}) + 1

= 3 (tan^{2} (\frac{π}{6}) + 1)

Use the following trivial identity:

tan (\frac{π}{6}) = \frac{3}{3}

tan (\frac{π}{6})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{3}{3}

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Cancel the common factor:

3

= \frac{1}{3}

= 3 (\frac{1}{3} + 1)

Join

\frac{1}{3} + 1 : \frac{4}{3}

\frac{1}{3} + 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 3 + 1 = 4

1 \cdot 3 + 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 + 1

Add the numbers:

3 + 1 = 4

= 4

= \frac{4}{3}

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

Multiply fractions:

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

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

Cancel the common factor:

3

= 4

= 4

cos (33 0^{\circ}) tan (- 31 5^{\circ}) - sin (- 21 0^{\circ}) cot (33 0^{\circ})

11 sin (\frac{π}{2} + 0.1)

2^{2} - 4.9 (2) + 5.7 cos (0.01 (2))

cos (30 \circ)

1 - 3 \cdot cos (\frac{π}{2})