sin(π2+x)=-tan(x)

Use the sum formula for sine to simplify the expression. The formula states that sin(A+B)=sin(A)cos(B)+cos(A)sin(B).

sin(π2)cos(x)+cos(π2)sin(x)=-tan(x)

Square both sides of the equation.

(sin(π2)cos(x)+cos(π2)sin(x))2=(-tan(x))2

The exact value of sin(π2) is 1.

(1cos(x)+cos(π2)sin(x))2=(-tan(x))2

Multiply cos(x) by 1.

(cos(x)+cos(π2)sin(x))2=(-tan(x))2

The exact value of cos(π2) is 0.

(cos(x)+0sin(x))2=(-tan(x))2

Multiply 0 by sin(x).

(cos(x)+0)2=(-tan(x))2

(cos(x)+0)2=(-tan(x))2

Add cos(x) and 0.

cos2(x)=(-tan(x))2

Apply the product rule to -tan(x).

cos2(x)=(-1)2tan2(x)

Raise -1 to the power of 2.

cos2(x)=1tan2(x)

Multiply tan2(x) by 1.

cos2(x)=tan2(x)

cos2(x)=tan2(x)

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=cos(x) and b=tan(x).

(cos(x)+tan(x))(cos(x)-tan(x))=0

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

cos(x)+tan(x)=0

cos(x)-tan(x)=0

Set the first factor equal to 0.

cos(x)+tan(x)=0

Rewrite tan(x) in terms of sines and cosines.

cos(x)+sin(x)cos(x)=0

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

1,cos(x),1

Since 1,cos(x),1 contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 1,1,1 then find LCM for the variable part cos1(x).

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number 1 is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.

1

The factor for cos1(x) is cos(x) itself.

cos1(x)=cos(x)

cos(x) occurs 1 time.

The LCM of cos1(x) is the result of multiplying all prime factors the greatest number of times they occur in either term.

cos(x)

cos(x)

Multiply each term by cos(x) and simplify.

Multiply each term in cos(x)+sin(x)cos(x)=0 by cos(x) in order to remove all the denominators from the equation.

cos(x)⋅cos(x)+sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Simplify each term.

Multiply cos(x) by cos(x).

cos2(x)+sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Cancel the common factor of cos(x).

Cancel the common factor.

cos2(x)+sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Rewrite the expression.

cos2(x)+sin(x)=0⋅cos(x)

cos2(x)+sin(x)=0⋅cos(x)

cos2(x)+sin(x)=0⋅cos(x)

Multiply 0 by cos(x).

cos2(x)+sin(x)=0

cos2(x)+sin(x)=0

Subtract sin(x) from both sides of the equation.

cos2(x)=-sin(x)

Take the square root of both sides of the equation to eliminate the exponent on the left side.

cos(x)=±-sin(x)

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the ± to find the first solution.

cos(x)=-sin(x)

Next, use the negative value of the ± to find the second solution.

cos(x)=–sin(x)

The complete solution is the result of both the positive and negative portions of the solution.

cos(x)=-sin(x),–sin(x)

cos(x)=-sin(x),–sin(x)

Set up each of the solutions to solve for x.

cos(x)=-sin(x)

cos(x)=–sin(x)

Set up the equation to solve for x.

cos(x)=-sin(x)

Solve the equation for x.

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

-sin(x)=cos(x)

To remove the radical on the left side of the equation, square both sides of the equation.

(-sin(x))2=cos2(x)

Simplify the left side of the equation.

-sin(x)=cos2(x)

Solve for x.

Move cos2(x) to the left side of the equation by subtracting it from both sides.

-sin(x)-cos2(x)=0

Replace the -cos2(x) with -(1-sin2(x)) based on the sin2(x)+cos2(x)=1 identity.

-sin(x)-(1-sin2(x))=0

Simplify each term.

Apply the distributive property.

-sin(x)-1⋅1+sin2(x)=0

Multiply -1 by 1.

-sin(x)-1+sin2(x)=0

Multiply –sin2(x).

Multiply -1 by -1.

-sin(x)-1+1sin2(x)=0

Multiply sin2(x) by 1.

-sin(x)-1+sin2(x)=0

-sin(x)-1+sin2(x)=0

-sin(x)-1+sin2(x)=0

Reorder the polynomial.

sin2(x)-sin(x)-1=0

Substitute u for sin(x).

(u)2-(u)-1=0

Use the quadratic formula to find the solutions.

-b±b2-4(ac)2a

Substitute the values a=1, b=-1, and c=-1 into the quadratic formula and solve for u.

1±(-1)2-4⋅(1⋅-1)2⋅1

Simplify.

Simplify the numerator.

Raise -1 to the power of 2.

u=1±1-4⋅(1⋅-1)2⋅1

Multiply -1 by 1.

u=1±1-4⋅-12⋅1

Multiply -4 by -1.

u=1±1+42⋅1

Add 1 and 4.

u=1±52⋅1

u=1±52⋅1

Multiply 2 by 1.

u=1±52

u=1±52

The final answer is the combination of both solutions.

u=1+52,1-52

Substitute sin(x) for u.

sin(x)=1+52,1-52

Set up each of the solutions to solve for x.

sin(x)=1+52

sin(x)=1-52

Set up the equation to solve for x.

sin(x)=1+52

The range of sine is -1≤y≤1. Since 1+52 does not fall in this range, there is no solution.

No solution

Set up the equation to solve for x.

sin(x)=1-52

Solve the equation for x.

Take the inverse sine of both sides of the equation to extract x from inside the sine.

x=arcsin(1-52)

Evaluate arcsin(1-52).

x=-0.66623943

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from π to find the solution in the second quadrant.

x=(3.14159265)+0.66623943

Simplify the expression to find the second solution.

Remove the parentheses around the expression 3.14159265.

x=3.14159265+0.66623943

Add 3.14159265 and 0.66623943.

x=3.80783208

x=3.80783208

Find the period.

The period of the function can be calculated using 2π|b|.

2π|b|

Replace b with 1 in the formula for period.

2π|1|

Solve the equation.

The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.

2π1

Divide 2π by 1.

2π

2π

2π

Add 2π to every negative angle to get positive angles.

Add 2π to -0.66623943 to find the positive angle.

-0.66623943+2π

Subtract 0.66623943 from 2π.

5.61694587

List the new angles.

x=5.61694587

x=5.61694587

The period of the sin(x) function is 2π so values will repeat every 2π radians in both directions.

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

List all of the results found in the previous steps.

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

Set up the equation to solve for x.

cos(x)=–sin(x)

Solve the equation for x.

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

–sin(x)=cos(x)

To remove the radical on the left side of the equation, square both sides of the equation.

(–sin(x))2=cos2(x)

Simplify the left side of the equation.

(-1)2(-sin(x))=cos2(x)

Solve for x.

Move cos2(x) to the left side of the equation by subtracting it from both sides.

(-1)2(-sin(x))-cos2(x)=0

Simplify each term.

Multiply (-1)2 by -1 by adding the exponents.

Move -1.

-(-1)2sin(x)-cos2(x)=0

Multiply -1 by (-1)2.

Raise -1 to the power of 1.

(-1)(-1)2sin(x)-cos2(x)=0

Use the power rule aman=am+n to combine exponents.

(-1)1+2sin(x)-cos2(x)=0

(-1)1+2sin(x)-cos2(x)=0

Add 1 and 2.

(-1)3sin(x)-cos2(x)=0

(-1)3sin(x)-cos2(x)=0

Raise -1 to the power of 3.

-1sin(x)-cos2(x)=0

Rewrite -1sin(x) as -sin(x).

-sin(x)-cos2(x)=0

-sin(x)-cos2(x)=0

Replace the -cos2(x) with -(1-sin2(x)) based on the sin2(x)+cos2(x)=1 identity.

-sin(x)-(1-sin2(x))=0

Simplify each term.

Apply the distributive property.

-sin(x)-1⋅1+sin2(x)=0

Multiply -1 by 1.

-sin(x)-1+sin2(x)=0

Multiply –sin2(x).

Multiply -1 by -1.

-sin(x)-1+1sin2(x)=0

Multiply sin2(x) by 1.

-sin(x)-1+sin2(x)=0

-sin(x)-1+sin2(x)=0

-sin(x)-1+sin2(x)=0

Reorder the polynomial.

sin2(x)-sin(x)-1=0

Substitute u for sin(x).

(u)2-(u)-1=0

Use the quadratic formula to find the solutions.

-b±b2-4(ac)2a

Substitute the values a=1, b=-1, and c=-1 into the quadratic formula and solve for u.

1±(-1)2-4⋅(1⋅-1)2⋅1

Simplify.

Simplify the numerator.

Raise -1 to the power of 2.

u=1±1-4⋅(1⋅-1)2⋅1

Multiply -1 by 1.

u=1±1-4⋅-12⋅1

Multiply -4 by -1.

u=1±1+42⋅1

Add 1 and 4.

u=1±52⋅1

u=1±52⋅1

Multiply 2 by 1.

u=1±52

u=1±52

The final answer is the combination of both solutions.

u=1+52,1-52

Substitute sin(x) for u.

sin(x)=1+52,1-52

Set up each of the solutions to solve for x.

sin(x)=1+52

sin(x)=1-52

Set up the equation to solve for x.

sin(x)=1+52

The range of sine is -1≤y≤1. Since 1+52 does not fall in this range, there is no solution.

No solution

Set up the equation to solve for x.

sin(x)=1-52

Solve the equation for x.

Take the inverse sine of both sides of the equation to extract x from inside the sine.

x=arcsin(1-52)

Evaluate arcsin(1-52).

x=-0.66623943

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from π to find the solution in the second quadrant.

x=(3.14159265)+0.66623943

Simplify the expression to find the second solution.

Remove the parentheses around the expression 3.14159265.

x=3.14159265+0.66623943

Add 3.14159265 and 0.66623943.

x=3.80783208

x=3.80783208

Find the period.

The period of the function can be calculated using 2π|b|.

2π|b|

Replace b with 1 in the formula for period.

2π|1|

Solve the equation.

The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.

2π1

Divide 2π by 1.

2π

2π

2π

Add 2π to every negative angle to get positive angles.

Add 2π to -0.66623943 to find the positive angle.

-0.66623943+2π

Subtract 0.66623943 from 2π.

5.61694587

List the new angles.

x=5.61694587

x=5.61694587

The period of the sin(x) function is 2π so values will repeat every 2π radians in both directions.

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

List all of the results found in the previous steps.

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn, for any integer n

The complete solution is the set of all solutions.

x=3.80783208+2πn,5.61694587+2πn,3.80783208+2πn,5.61694587+2πn, for any integer n

x=3.80783208+2πn,5.61694587+2πn,3.80783208+2πn,5.61694587+2πn, for any integer n

Set the next factor equal to 0.

cos(x)-tan(x)=0

Rewrite tan(x) in terms of sines and cosines.

cos(x)-sin(x)cos(x)=0

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

1,cos(x),1

Since 1,cos(x),1 contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part 1,1,1 then find LCM for the variable part cos1(x).

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number 1 is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.

1

The factor for cos1(x) is cos(x) itself.

cos1(x)=cos(x)

cos(x) occurs 1 time.

The LCM of cos1(x) is the result of multiplying all prime factors the greatest number of times they occur in either term.

cos(x)

cos(x)

Multiply each term by cos(x) and simplify.

Multiply each term in cos(x)-sin(x)cos(x)=0 by cos(x) in order to remove all the denominators from the equation.

cos(x)⋅cos(x)-sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Simplify each term.

Multiply cos(x) by cos(x).

cos2(x)-sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Cancel the common factor of cos(x).

Move the leading negative in -sin(x)cos(x) into the numerator.

cos2(x)+-sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Cancel the common factor.

cos2(x)+-sin(x)cos(x)⋅cos(x)=0⋅cos(x)

Rewrite the expression.

cos2(x)-sin(x)=0⋅cos(x)

cos2(x)-sin(x)=0⋅cos(x)

cos2(x)-sin(x)=0⋅cos(x)

Multiply 0 by cos(x).

cos2(x)-sin(x)=0

cos2(x)-sin(x)=0

Add sin(x) to both sides of the equation.

cos2(x)=sin(x)

Take the square root of both sides of the equation to eliminate the exponent on the left side.

cos(x)=±sin(x)

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the ± to find the first solution.

cos(x)=sin(x)

Next, use the negative value of the ± to find the second solution.

cos(x)=-sin(x)

The complete solution is the result of both the positive and negative portions of the solution.

cos(x)=sin(x),-sin(x)

cos(x)=sin(x),-sin(x)

Set up each of the solutions to solve for x.

cos(x)=sin(x)

cos(x)=-sin(x)

Set up the equation to solve for x.

cos(x)=sin(x)

Solve the equation for x.

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

sin(x)=cos(x)

To remove the radical on the left side of the equation, square both sides of the equation.

(sin(x))2=cos2(x)

Simplify the left side of the equation.

sin(x)=cos2(x)

Solve for x.

Move cos2(x) to the left side of the equation by subtracting it from both sides.

sin(x)-cos2(x)=0

Replace the -cos2(x) with -(1-sin2(x)) based on the sin2(x)+cos2(x)=1 identity.

sin(x)-(1-sin2(x))=0

Simplify each term.

Apply the distributive property.

sin(x)-1⋅1+sin2(x)=0

Multiply -1 by 1.

sin(x)-1+sin2(x)=0

Multiply –sin2(x).

Multiply -1 by -1.

sin(x)-1+1sin2(x)=0

Multiply sin2(x) by 1.

sin(x)-1+sin2(x)=0

sin(x)-1+sin2(x)=0

sin(x)-1+sin2(x)=0

Reorder the polynomial.

sin2(x)+sin(x)-1=0

Substitute u for sin(x).

(u)2+u-1=0

Use the quadratic formula to find the solutions.

-b±b2-4(ac)2a

Substitute the values a=1, b=1, and c=-1 into the quadratic formula and solve for u.

-1±12-4⋅(1⋅-1)2⋅1

Simplify.

Simplify the numerator.

One to any power is one.

u=-1±1-4⋅(1⋅-1)2⋅1

Multiply -1 by 1.

u=-1±1-4⋅-12⋅1

Multiply -4 by -1.

u=-1±1+42⋅1

Add 1 and 4.

u=-1±52⋅1

u=-1±52⋅1

Multiply 2 by 1.

u=-1±52

Factor -1 out of -1±5.

u=-11±52

Multiply -1 by -1.

u=1-1±52

Multiply -1±5 by 1.

u=-1±52

u=-1±52

The final answer is the combination of both solutions.

u=-1-52,-1+52

Substitute sin(x) for u.

sin(x)=-1-52,-1+52

Set up each of the solutions to solve for x.

sin(x)=-1-52

sin(x)=-1+52

Set up the equation to solve for x.

sin(x)=-1-52

Solve the equation for x.

Take the inverse sine of both sides of the equation to extract x from inside the sine.

x=arcsin(-1-52)

Evaluate arcsin(-1-52).

x=0.66623943

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from 2π, to find a reference angle. Next, add this reference angle to π to find the solution in the third quadrant.

x=2(3.14159265)-0.66623943+3.14159265

Simplify the expression to find the second solution.

Remove the parentheses around the expression 3.14159265.

x=2(3.14159265)-0.66623943+3.14159265

Simplify 2(3.14159265)-0.66623943+3.14159265.

Multiply 2 by 3.14159265.

x=6.2831853-0.66623943+3.14159265

Subtract 0.66623943 from 6.2831853.

x=5.61694587+3.14159265

Add 5.61694587 and 3.14159265.

x=8.75853852

x=8.75853852

x=8.75853852

Find the period.

The period of the function can be calculated using 2π|b|.

2π|b|

Replace b with 1 in the formula for period.

2π|1|

Solve the equation.

The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.

2π1

Divide 2π by 1.

2π

2π

2π

The period of the sin(x) function is 2π so values will repeat every 2π radians in both directions.

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

Set up the equation to solve for x.

sin(x)=-1+52

The range of sine is -1≤y≤1. Since -1+52 does not fall in this range, there is no solution.

No solution

List all of the results found in the previous steps.

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

Set up the equation to solve for x.

cos(x)=-sin(x)

Solve the equation for x.

-sin(x)=cos(x)

To remove the radical on the left side of the equation, square both sides of the equation.

(-sin(x))2=cos2(x)

Simplify the left side of the equation.

(-1)2(sin(x))=cos2(x)

Solve for x.

Move cos2(x) to the left side of the equation by subtracting it from both sides.

(-1)2(sin(x))-cos2(x)=0

Simplify each term.

Raise -1 to the power of 2.

1sin(x)-cos2(x)=0

Multiply sin(x) by 1.

sin(x)-cos2(x)=0

sin(x)-cos2(x)=0

Replace the -cos2(x) with -(1-sin2(x)) based on the sin2(x)+cos2(x)=1 identity.

sin(x)-(1-sin2(x))=0

Simplify each term.

Apply the distributive property.

sin(x)-1⋅1+sin2(x)=0

Multiply -1 by 1.

sin(x)-1+sin2(x)=0

Multiply –sin2(x).

Multiply -1 by -1.

sin(x)-1+1sin2(x)=0

Multiply sin2(x) by 1.

sin(x)-1+sin2(x)=0

sin(x)-1+sin2(x)=0

sin(x)-1+sin2(x)=0

Reorder the polynomial.

sin2(x)+sin(x)-1=0

Substitute u for sin(x).

(u)2+u-1=0

Use the quadratic formula to find the solutions.

-b±b2-4(ac)2a

Substitute the values a=1, b=1, and c=-1 into the quadratic formula and solve for u.

-1±12-4⋅(1⋅-1)2⋅1

Simplify.

Simplify the numerator.

One to any power is one.

u=-1±1-4⋅(1⋅-1)2⋅1

Multiply -1 by 1.

u=-1±1-4⋅-12⋅1

Multiply -4 by -1.

u=-1±1+42⋅1

Add 1 and 4.

u=-1±52⋅1

u=-1±52⋅1

Multiply 2 by 1.

u=-1±52

Factor -1 out of -1±5.

u=-11±52

Multiply -1 by -1.

u=1-1±52

Multiply -1±5 by 1.

u=-1±52

u=-1±52

The final answer is the combination of both solutions.

u=-1-52,-1+52

Substitute sin(x) for u.

sin(x)=-1-52,-1+52

Set up each of the solutions to solve for x.

sin(x)=-1-52

sin(x)=-1+52

Set up the equation to solve for x.

sin(x)=-1-52

Solve the equation for x.

Take the inverse sine of both sides of the equation to extract x from inside the sine.

x=arcsin(-1-52)

Evaluate arcsin(-1-52).

x=0.66623943

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from 2π, to find a reference angle. Next, add this reference angle to π to find the solution in the third quadrant.

x=2(3.14159265)-0.66623943+3.14159265

Simplify the expression to find the second solution.

Remove the parentheses around the expression 3.14159265.

x=2(3.14159265)-0.66623943+3.14159265

Simplify 2(3.14159265)-0.66623943+3.14159265.

Multiply 2 by 3.14159265.

x=6.2831853-0.66623943+3.14159265

Subtract 0.66623943 from 6.2831853.

x=5.61694587+3.14159265

Add 5.61694587 and 3.14159265.

x=8.75853852

x=8.75853852

x=8.75853852

Find the period.

The period of the function can be calculated using 2π|b|.

2π|b|

Replace b with 1 in the formula for period.

2π|1|

Solve the equation.

The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.

2π1

Divide 2π by 1.

2π

2π

2π

The period of the sin(x) function is 2π so values will repeat every 2π radians in both directions.

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

Set up the equation to solve for x.

sin(x)=-1+52

The range of sine is -1≤y≤1. Since -1+52 does not fall in this range, there is no solution.

No solution

List all of the results found in the previous steps.

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn, for any integer n

The complete solution is the set of all solutions.

x=0.66623943+2πn,8.75853852+2πn,0.66623943+2πn,8.75853852+2πn, for any integer n

x=0.66623943+2πn,8.75853852+2πn,0.66623943+2πn,8.75853852+2πn, for any integer n

The final solution is all the values that make (cos(x)+tan(x))(cos(x)-tan(x))=0 true.

x=3.80783208+2πn,5.61694587+2πn,0.66623943+2πn,8.75853852+2πn, for any integer n

Consolidate 3.80783208+2πn and 0.66623943+2πn to 0.66623943+πn.

x=0.66623943+πn,5.61694587+2πn,8.75853852+2πn, for any integer n

Exclude the solutions that do not make sin(π2+x)=-tan(x) true.

x=3.80783208,5.61694587,10.09101739,11.90013118,16.3742027,18.18331648,22.657388,24.46650179,30.7496871,37.03287241,43.31605771,49.59924302, for any integer n

Solve for ? sin(pi/2+x)=-tan(x)