What is the general solution for tan^2(θ)= 3/8 ?

The general solution for tan^2(θ)= 3/8 is θ=0.54946…+pin,θ=-0.54946…+pin

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tan^2(θ)= 3/8

Solution

tan^{2} (θ) = \frac{3}{8}

Solution

θ = 0.54946 \dots + πn, θ = - 0.54946 \dots + πn

Degrees

θ = 31.48215 \dots^{\circ} + 18 0^{\circ} n, θ = - 31.48215 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (θ) = \frac{3}{8}

Solve by substitution

tan^{2} (θ) = \frac{3}{8}

Let:

tan (θ) = u

u^{2} = \frac{3}{8}

u^{2} = \frac{3}{8} : u = \frac{6}{4}, u = - \frac{6}{4}

u^{2} = \frac{3}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3}{8}, u = - \frac{3}{8}

\frac{3}{8} = \frac{6}{4}

\frac{3}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{3}{2 2}

Rationalize

\frac{3}{2 2} : \frac{6}{4}

\frac{3}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{6}{4}

= \frac{6}{4}

- \frac{3}{8} = - \frac{6}{4}

- \frac{3}{8}

Simplify

\frac{3}{8} : \frac{3}{2 2}

\frac{3}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{3}{2 2}

= - \frac{3}{2 2}

Rationalize

- \frac{3}{2 2} : - \frac{6}{4}

- \frac{3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{3 2}{2 2 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{6}{4}

= - \frac{6}{4}

u = \frac{6}{4}, u = - \frac{6}{4}

Substitute back

u = tan (θ)

tan (θ) = \frac{6}{4}, tan (θ) = - \frac{6}{4}

tan (θ) = \frac{6}{4}, tan (θ) = - \frac{6}{4}

tan (θ) = \frac{6}{4} : θ = arctan (\frac{6}{4}) + πn

tan (θ) = \frac{6}{4}

Apply trig inverse properties

tan (θ) = \frac{6}{4}

General solutions for

tan (θ) = \frac{6}{4}

tan (x) = a \Rightarrow x = arctan (a) + πn

θ = arctan (\frac{6}{4}) + πn

θ = arctan (\frac{6}{4}) + πn

tan (θ) = - \frac{6}{4} : θ = arctan (- \frac{6}{4}) + πn

tan (θ) = - \frac{6}{4}

Apply trig inverse properties

tan (θ) = - \frac{6}{4}

General solutions for

tan (θ) = - \frac{6}{4}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

θ = arctan (- \frac{6}{4}) + πn

θ = arctan (- \frac{6}{4}) + πn

Combine all the solutions

θ = arctan (\frac{6}{4}) + πn, θ = arctan (- \frac{6}{4}) + πn

Show solutions in decimal form

θ = 0.54946 \dots + πn, θ = - 0.54946 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(θ)= 3/8 ?
The general solution for tan^2(θ)= 3/8 is θ=0.54946…+pin,θ=-0.54946…+pin