Trigonometry Examples

$(\cos (- t)) = - \frac{1}{4}$

Step 1

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

$- t = \arccos (- \frac{1}{4})$

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Evaluate $\arccos (- \frac{1}{4})$ .

$- t = 1.82347658$

Step 3

Divide each term in $- t = 1.82347658$ by $- 1$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $- t = 1.82347658$ by $- 1$ .

$\frac{- t}{- 1} = \frac{1.82347658}{- 1}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{1.82347658}{- 1}$

Step 3.2.2

Divide $t$ by $1$ .

$t = \frac{1.82347658}{- 1}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Divide $1.82347658$ by $- 1$ .

$t = - 1.82347658$

Step 4

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

$- t = 2 (3.14159265) - 1.82347658$

Step 5

Divide each term in $- t = 2 (3.14159265) - 1.82347658$ by $- 1$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $- t = 2 (3.14159265) - 1.82347658$ by $- 1$ .

$\frac{- t}{- 1} = \frac{2 (3.14159265)}{- 1} + \frac{- 1.82347658}{- 1}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{2 (3.14159265)}{- 1} + \frac{- 1.82347658}{- 1}$

Step 5.2.2

Divide $t$ by $1$ .

$t = \frac{2 (3.14159265)}{- 1} + \frac{- 1.82347658}{- 1}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Combine the numerators over the common denominator.

$t = \frac{2 (3.14159265) - 1.82347658}{- 1}$

Step 5.3.2

Simplify the expression.

Tap for more steps...

Step 5.3.2.1

Multiply $2$ by $3.14159265$ .

$t = \frac{6.2831853 - 1.82347658}{- 1}$

Step 5.3.2.2

Subtract $1.82347658$ from $6.2831853$ .

$t = \frac{4.45970872}{- 1}$

Step 5.3.2.3

Divide $4.45970872$ by $- 1$ .

$t = - 4.45970872$

Step 6

Find the period of $\cos (- t)$ .

Tap for more steps...

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 $- 1$ and $0$ 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.

Tap for more steps...

Step 7.1

Add $2 π$ to $- 1.82347658$ to find the positive angle.

$- 1.82347658 + 2 π$

Step 7.2

Subtract $1.82347658$ from $2 π$ .

$4.45970872$

Step 7.3

Add $2 π$ to $- 4.45970872$ to find the positive angle.

$- 4.45970872 + 2 π$

Step 7.4

Subtract $4.45970872$ from $2 π$ .

$1.82347658$

Step 7.5

List the new angles.

$t = 1.82347658$

Step 8

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

$t = 1.82347658 + 2 π n, 4.45970872 + 2 π n$ , for any integer $n$