What is the value of tan^2(30)+csc^2(60) ?

The value of tan^2(30)+csc^2(60) is 5/3

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Popular >

tan^2(30)+csc^2(60)

Solution

tan^{2} (3 0^{\circ}) + csc^{2} (6 0^{\circ})

Solution

\frac{5}{3}

Decimal Notation

1.66666 \dots

Solution steps

tan^{2} (3 0^{\circ}) + csc^{2} (6 0^{\circ})

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:

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

csc (6 0^{\circ})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

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

Simplify

(\frac{3}{3})^{2} + (\frac{2 3}{3})^{2} : \frac{5}{3}

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

(\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}

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

(\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}

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

Apply rule

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

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

1 + 2^{2} = 5

1 + 2^{2}

2^{2} = 4

= 1 + 4

Add the numbers:

1 + 4 = 5

= 5

= \frac{5}{3}

= \frac{5}{3}

Popular Examples

10*csc(23)

10 \cdot csc (2 3^{\circ})

arcsin(sin((5pi)/(12)))

arcsin (sin (\frac{5 π}{12}))

(cos((2pi)/3)+isin((2pi)/3))^2

(cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3}))^{2}

csc(255)

csc (25 5^{\circ})

1/(tan(20))

\frac{1}{tan ( 2 0 ^{\circ} )}

Frequently Asked Questions (FAQ)

What is the value of tan^2(30)+csc^2(60) ?
The value of tan^2(30)+csc^2(60) is 5/3