What is the general solution for tan^2(x)=500 ?

The general solution for tan^2(x)=500 is x=1.52610…+pin,x=-1.52610…+pin

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Popular Trigonometry >

tan^2(x)=500

Solution

tan^{2} (x) = 500

Solution

x = 1.52610 \dots + πn, x = - 1.52610 \dots + πn

Degrees

x = 87.43936 \dots^{\circ} + 18 0^{\circ} n, x = - 87.43936 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) = 500

Solve by substitution

tan^{2} (x) = 500

Let:

tan (x) = u

u^{2} = 500

u^{2} = 500 : u = 105, u = - 105

u^{2} = 500

For

x^{2} = f (a)

the solutions are

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

u = 500, u = - 500

500 = 105

500

Prime factorization of

500 : 2^{2} \cdot 5^{3}

500

500

divides by

2500 = 250 \cdot 2

= 2 \cdot 250

250

divides by

2250 = 125 \cdot 2

= 2 \cdot 2 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5

= 2^{2} \cdot 5^{3}

= 5^{3} \cdot 2^{2}

Apply exponent rule:

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

= 2^{2} \cdot 5^{2} \cdot 5

Apply radical rule:

= 5 2^{2} 5^{2}

Apply radical rule:

2^{2} = 2

= 25 5^{2}

Apply radical rule:

5^{2} = 5

= 2 \cdot 55

Refine

= 105

- 500 = - 105

- 500

500 = 105

500

Prime factorization of

500 : 2^{2} \cdot 5^{3}

500

500

divides by

2500 = 250 \cdot 2

= 2 \cdot 250

250

divides by

2250 = 125 \cdot 2

= 2 \cdot 2 \cdot 125

125

divides by

5125 = 25 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5

= 2^{2} \cdot 5^{3}

= 5^{3} \cdot 2^{2}

Apply exponent rule:

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

= 2^{2} \cdot 5^{2} \cdot 5

Apply radical rule:

= 5 2^{2} 5^{2}

Apply radical rule:

2^{2} = 2

= 25 5^{2}

Apply radical rule:

5^{2} = 5

= 2 \cdot 55

Refine

= 105

= - 105

u = 105, u = - 105

Substitute back

u = tan (x)

tan (x) = 105, tan (x) = - 105

tan (x) = 105, tan (x) = - 105

tan (x) = 105 : x = arctan (105) + πn

tan (x) = 105

Apply trig inverse properties

tan (x) = 105

General solutions for

tan (x) = 105

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

x = arctan (105) + πn

x = arctan (105) + πn

tan (x) = - 105 : x = arctan (- 105) + πn

tan (x) = - 105

Apply trig inverse properties

tan (x) = - 105

General solutions for

tan (x) = - 105

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

x = arctan (- 105) + πn

x = arctan (- 105) + πn

Combine all the solutions

x = arctan (105) + πn, x = arctan (- 105) + πn

Show solutions in decimal form

x = 1.52610 \dots + πn, x = - 1.52610 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)=500 ?
The general solution for tan^2(x)=500 is x=1.52610…+pin,x=-1.52610…+pin