Trigonometry Examples

$\cos (a + 60) = \frac{4}{5}$

Step 1

Take the inverse cosine of both sides of the equation to extract $a$ from inside the cosine.

$a + 60 = \arccos (\frac{4}{5})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arccos (\frac{4}{5})$ .

$a + 60 = 0.6435011$

Step 3

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1

Subtract $60$ from both sides of the equation.

$a = 0.6435011 - 60$

Step 3.2

Subtract $60$ from $0.6435011$ .

$a = - 59.35649889$

Step 4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$a + 60 = 2 (3.14159265) - 0.6435011$

Step 5

Solve for $a$ .

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Step 5.1

Simplify $2 (3.14159265) - 0.6435011$ .

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Step 5.1.1

Multiply $2$ by $3.14159265$ .

$a + 60 = 6.2831853 - 0.6435011$

Step 5.1.2

Subtract $0.6435011$ from $6.2831853$ .

$a + 60 = 5.63968419$

Step 5.2

Move all terms not containing $a$ to the right side of the equation.

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Step 5.2.1

Subtract $60$ from both sides of the equation.

$a = 5.63968419 - 60$

Step 5.2.2

Subtract $60$ from $5.63968419$ .

$a = - 54.3603158$

Step 6

Find the period of $\cos (a + 60)$ .

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Step 6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

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Step 7.1

Add $2 π$ to $- 59.35649889$ to find the positive angle.

$- 59.35649889 + 2 π$

Step 7.2

Subtract $59.35649889$ from $2 π$ .

$- 53.07331358$

Step 7.3

Add $2 π$ to $- 54.3603158$ to find the positive angle.

$- 54.3603158 + 2 π$

Step 7.4

Subtract $54.3603158$ from $2 π$ .

$- 48.07713049$

Step 7.5

List the new angles.

$a = - 48.07713049$

Step 8

The period of the $\cos (a + 60)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$a = - 53.07331358 + 2 π n, - 48.07713049 + 2 π n$ , for any integer $n$