What is the value of sec^2(150) ?

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sec^2(150)

sec^{2} (15 0^{\circ})

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

sec^{2} (15 0^{\circ})

Use the following trivial identity:

sec (15 0^{\circ}) = - \frac{2 3}{3}

sec (15 0^{\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}

= - \frac{2 3}{3}

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

Simplify

(- \frac{2 3}{3})^{2} : \frac{4}{3}

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(23)^{2} = 2^{2} (3)^{2}

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

Cancel the common factor:

3

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

2^{2} = 4

= \frac{4}{3}

= \frac{4}{3}

- tan (\frac{π}{3})

arctan (\frac{29}{38})

tan (9 0^{\circ} + 6 0^{\circ})

2 cos^{2} (π)

arctan (\frac{\frac{1}{2}}{\frac{1}{2}})