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Popular Trigonometry >

75cos^2(x)-34cos(x)+3=0

Solution

75 cos^{2} (x) - 34 cos (x) + 3 = 0

Solution

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

Degrees

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n, x = 83.10789 \dots^{\circ} + 36 0^{\circ} n, x = 276.89210 \dots^{\circ} + 36 0^{\circ} n

Solution steps

75 cos^{2} (x) - 34 cos (x) + 3 = 0

Solve by substitution

75 cos^{2} (x) - 34 cos (x) + 3 = 0

Let:

cos (x) = u

75 u^{2} - 34 u + 3 = 0

75 u^{2} - 34 u + 3 = 0 : u = \frac{1}{3}, u = \frac{3}{25}

75 u^{2} - 34 u + 3 = 0

Solve with the quadratic formula

75 u^{2} - 34 u + 3 = 0

Quadratic Equation Formula:

For

a = 75, b = - 34, c = 3

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

(- 34)^{2} - 4 \cdot 75 \cdot 3 = 16

(- 34)^{2} - 4 \cdot 75 \cdot 3

Apply exponent rule:

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

n

is even

(- 34)^{2} = 3 4^{2}

= 3 4^{2} - 4 \cdot 75 \cdot 3

Multiply the numbers:

4 \cdot 75 \cdot 3 = 900

= 3 4^{2} - 900

3 4^{2} = 1156

= 1156 - 900

Subtract the numbers:

1156 - 900 = 256

= 256

Factor the number:

256 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- ( - 34 ) \pm 16}{2 \cdot 75}

Separate the solutions

u_{1} = \frac{- ( - 34 ) + 16}{2 \cdot 75}, u_{2} = \frac{- ( - 34 ) - 16}{2 \cdot 75}

u = \frac{- ( - 34 ) + 16}{2 \cdot 75} : \frac{1}{3}

\frac{- ( - 34 ) + 16}{2 \cdot 75}

Apply rule

- (- a) = a

= \frac{34 + 16}{2 \cdot 75}

Add the numbers:

34 + 16 = 50

= \frac{50}{2 \cdot 75}

Multiply the numbers:

2 \cdot 75 = 150

= \frac{50}{150}

Cancel the common factor:

50

= \frac{1}{3}

u = \frac{- ( - 34 ) - 16}{2 \cdot 75} : \frac{3}{25}

\frac{- ( - 34 ) - 16}{2 \cdot 75}

Apply rule

- (- a) = a

= \frac{34 - 16}{2 \cdot 75}

Subtract the numbers:

34 - 16 = 18

= \frac{18}{2 \cdot 75}

Multiply the numbers:

2 \cdot 75 = 150

= \frac{18}{150}

Cancel the common factor:

6

= \frac{3}{25}

The solutions to the quadratic equation are:

u = \frac{1}{3}, u = \frac{3}{25}

Substitute back

u = cos (x)

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

cos (x) = \frac{1}{3} : x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = \frac{1}{3}

Apply trig inverse properties

cos (x) = \frac{1}{3}

General solutions for

cos (x) = \frac{1}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = \frac{3}{25} : x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

cos (x) = \frac{3}{25}

Apply trig inverse properties

cos (x) = \frac{3}{25}

General solutions for

cos (x) = \frac{3}{25}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

Combine all the solutions

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn, x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

Show solutions in decimal form

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

Graph

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