Algebra
Calculus
Trigonometry
Matrix
Question
Solve the equation
x=⎩⎨⎧−arccos(4−1+17)+2kπarccos(4−1+17)+2kπ,k∈Z
Alternative Form
x≈{−38.668282∘+360∘k38.668282∘+360∘k,k∈Z
Alternative Form
x≈{−0.674889+2kπ0.674889+2kπ,k∈Z
Evaluate
4sin2(x)=2cos(x)×1
Multiply the terms
4sin2(x)=2cos(x)
Use sin2(x)=1−cos2(x) to rewrite the expression
4−4cos2(x)=2cos(x)
Move the expression to the left side
4−4cos2(x)−2cos(x)=0
Rewrite in standard form
−4cos2(x)−2cos(x)+4=0
Multiply both sides
4cos2(x)+2cos(x)−4=0
Substitute a=4,b=2 and c=−4 into the quadratic formula cos(x)=2a−b±b2−4ac
cos(x)=2×4−2±22−4×4(−4)
Simplify the expression
cos(x)=8−2±22−4×4(−4)
Simplify the expression
More Steps
Evaluate
22−4×4(−4)
Multiply
More Steps
Multiply the terms
4×4(−4)
Rewrite the expression
−4×4×4
Multiply the terms with the same base by adding their exponents
−41+1+1
Add the numbers
−43
22−(−43)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
22+43
Evaluate the power
4+43
Evaluate the power
4+64
Add the numbers
68
cos(x)=8−2±68
Simplify the radical expression
More Steps
Evaluate
68
Write the expression as a product where the root of one of the factors can be evaluated
4×17
Write the number in exponential form with the base of 2
22×17
The root of a product is equal to the product of the roots of each factor
22×17
Reduce the index of the radical and exponent with 2
217
cos(x)=8−2±217
Separate the equation into 2 possible cases
cos(x)=8−2+217cos(x)=8−2−217
Simplify the expression
More Steps
Evaluate
cos(x)=8−2+217
Divide the terms
More Steps
Evaluate
8−2+217
Rewrite the expression
82(−1+17)
Cancel out the common factor 2
4−1+17
cos(x)=4−1+17
cos(x)=4−1+17cos(x)=8−2−217
Simplify the expression
More Steps
Evaluate
cos(x)=8−2−217
Divide the terms
More Steps
Evaluate
8−2−217
Rewrite the expression
82(−1−17)
Cancel out the common factor 2
4−1−17
Use b−a=−ba=−ba to rewrite the fraction
−41+17
cos(x)=−41+17
cos(x)=4−1+17cos(x)=−41+17
Rearrange the terms
cos(x)=4−1+17x∈/R
Calculate
More Steps
Evaluate
cos(x)=4−1+17
Use the inverse trigonometric function
x=arccos(4−1+17)
Calculate
x=−arccos(4−1+17)x=arccos(4−1+17)
Add the period of 2kπ,k∈Z to find all solutions
x=−arccos(4−1+17)+2kπ,k∈Zx=arccos(4−1+17)+2kπ,k∈Z
Find the union
x=⎩⎨⎧−arccos(4−1+17)+2kπarccos(4−1+17)+2kπ,k∈Z
x=⎩⎨⎧−arccos(4−1+17)+2kπarccos(4−1+17)+2kπ,k∈Zx∈/R
Solution
x=⎩⎨⎧−arccos(4−1+17)+2kπarccos(4−1+17)+2kπ,k∈Z
Alternative Form
x≈{−38.668282∘+360∘k38.668282∘+360∘k,k∈Z
Alternative Form
x≈{−0.674889+2kπ0.674889+2kπ,k∈Z
Show Solution
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