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Precalculus Examples
cos2(x)−2.4cos(x)−0.81=0
Step 1
Substitute u for cos(x).
(u)2−2.4u−0.81=0
Step 2
Use the quadratic formula to find the solutions.
−b±√b2−4(ac)2a
Step 3
Substitute the values a=1, b=−2.4, and c=−0.81 into the quadratic formula and solve for u.
2.4±√(−2.4)2−4⋅(1⋅−0.81)2⋅1
Step 4
Step 4.1
Simplify the numerator.
Step 4.1.1
Raise −2.4 to the power of 2.
u=2.4±√5.76−4⋅1⋅−0.812⋅1
Step 4.1.2
Multiply −4⋅1⋅−0.81.
Step 4.1.2.1
Multiply −4 by 1.
u=2.4±√5.76−4⋅−0.812⋅1
Step 4.1.2.2
Multiply −4 by −0.81.
u=2.4±√5.76+3.242⋅1
u=2.4±√5.76+3.242⋅1
Step 4.1.3
Add 5.76 and 3.24.
u=2.4±√92⋅1
Step 4.1.4
Rewrite 9 as 32.
u=2.4±√322⋅1
Step 4.1.5
Pull terms out from under the radical, assuming positive real numbers.
u=2.4±32⋅1
u=2.4±32⋅1
Step 4.2
Multiply 2 by 1.
u=2.4±32
u=2.4±32
Step 5
The final answer is the combination of both solutions.
u=2.7,−0.3
Step 6
Substitute cos(x) for u.
cos(x)=2.7,−0.3
Step 7
Set up each of the solutions to solve for x.
cos(x)=2.7
cos(x)=−0.3
Step 8
Step 8.1
The range of cosine is −1≤y≤1. Since 2.7 does not fall in this range, there is no solution.
No solution
No solution
Step 9
Step 9.1
Take the inverse cosine of both sides of the equation to extract x from inside the cosine.
x=arccos(−0.3)
Step 9.2
Simplify the right side.
Step 9.2.1
Evaluate arccos(−0.3).
x=1.87548898
x=1.87548898
Step 9.3
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from 2π to find the solution in the third quadrant.
x=2(3.14159265)−1.87548898
Step 9.4
Solve for x.
Step 9.4.1
Remove parentheses.
x=2(3.14159265)−1.87548898
Step 9.4.2
Simplify 2(3.14159265)−1.87548898.
Step 9.4.2.1
Multiply 2 by 3.14159265.
x=6.2831853−1.87548898
Step 9.4.2.2
Subtract 1.87548898 from 6.2831853.
x=4.40769632
x=4.40769632
x=4.40769632
Step 9.5
Find the period of cos(x).
Step 9.5.1
The period of the function can be calculated using 2π|b|.
2π|b|
Step 9.5.2
Replace b with 1 in the formula for period.
2π|1|
Step 9.5.3
The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.
2π1
Step 9.5.4
Divide 2π by 1.
2π
2π
Step 9.6
The period of the cos(x) function is 2π so values will repeat every 2π radians in both directions.
x=1.87548898+2πn,4.40769632+2πn, for any integer n
x=1.87548898+2πn,4.40769632+2πn, for any integer n
Step 10
List all of the solutions.
x=1.87548898+2πn,4.40769632+2πn, for any integer n