What is the general solution for 4cos(x-45)+7=10,0<= x<= 360 ?

Q: What is the general solution for 4cos(x-45)+7=10,0<= x<= 360 ?

The general solution for 4cos(x-45)+7=10,0<= x<= 360 is x=(180+4*0.72273…)/4 ,x=(180-4*0.72273…)/4

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4cos(x-45)+7=10,0<= x<= 360

Solution

4 cos (x - 4 5^{\circ}) + 7 = 10, 0^{\circ} \leq x \leq 36 0^{\circ}

Solution

x = \frac{18 0 ^{\circ} + 4 \cdot 0.72273 \dots}{4}, x = \frac{18 0 ^{\circ} - 4 \cdot 0.72273 \dots}{4}

Radians

x = \frac{π + 4 \cdot 0.72273 \dots}{4}, x = \frac{π - 4 \cdot 0.72273 \dots}{4}

Solution steps

4 cos (x - 4 5^{\circ}) + 7 = 10, 0^{\circ} \leq x \leq 36 0^{\circ}

Move

7

to the right side

4 cos (x - 4 5^{\circ}) + 7 = 10

Subtract

7

from both sides

4 cos (x - 4 5^{\circ}) + 7 - 7 = 10 - 7

Simplify

4 cos (x - 4 5^{\circ}) = 3

4 cos (x - 4 5^{\circ}) = 3

Divide both sides by

4

4 cos (x - 4 5^{\circ}) = 3

Divide both sides by

4

\frac{4 cos ( x - 4 5 ^{\circ} )}{4} = \frac{3}{4}

Simplify

cos (x - 4 5^{\circ}) = \frac{3}{4}

cos (x - 4 5^{\circ}) = \frac{3}{4}

Apply trig inverse properties

cos (x - 4 5^{\circ}) = \frac{3}{4}

General solutions for

cos (x - 4 5^{\circ}) = \frac{3}{4}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x - 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n, x - 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

x - 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n, x - 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

Solve

x - 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n : x = arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

x - 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} = arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x = arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

x = arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

Solve

x - 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n : x = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

x - 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

x = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

x = arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}, x = 36 0^{\circ} - arccos (\frac{3}{4}) + 36 0^{\circ} n + 4 5^{\circ}

Solutions for the range

0 \leq x \leq 36 0^{\circ}

x = \frac{18 0 ^{\circ} + 4 arccos ( \frac{3}{4} )}{4}, x = \frac{18 0 ^{\circ} - 4 arccos ( \frac{3}{4} )}{4}

Show solutions in decimal form

x = \frac{18 0 ^{\circ} + 4 \cdot 0.72273 \dots}{4}, x = \frac{18 0 ^{\circ} - 4 \cdot 0.72273 \dots}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos(x-45)+7=10,0<= x<= 360 ?
The general solution for 4cos(x-45)+7=10,0<= x<= 360 is x=(180+4*0.72273…)/4 ,x=(180-4*0.72273…)/4