Trigonometry Examples

$\sin^{2} (θ) + 3 \sin (θ) - 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \sin (θ)$ . Substitute $u$ for all occurrences of $\sin (θ)$ .

$u^{2} + 3 u - 4 = 0$

Step 1.2

Factor $u^{2} + 3 u - 4$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2.2

Write the factored form using these integers.

$(u - 1) (u + 4) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sin (θ)$ .

$(\sin (θ) - 1) (\sin (θ) + 4) = 0$

Step 2

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

$\sin (θ) - 1 = 0$

$\sin (θ) + 4 = 0$

Step 3

Set $\sin (θ) - 1$ equal to $0$ and solve for $θ$ .

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Step 3.1

Set $\sin (θ) - 1$ equal to $0$ .

$\sin (θ) - 1 = 0$

Step 3.2

Solve $\sin (θ) - 1 = 0$ for $θ$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$\sin (θ) = 1$

Step 3.2.2

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

$θ = \arcsin (1)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

The exact value of $\arcsin (1)$ is $90$ .

$θ = 90$

Step 3.2.4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 90$

Step 3.2.5

Subtract $90$ from $180$ .

$θ = 90$

Step 3.2.6

Find the period of $\sin (θ)$ .

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Step 3.2.6.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 3.2.6.2

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

$\frac{360}{| 1 |}$

Step 3.2.6.3

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

$\frac{360}{1}$

Step 3.2.6.4

Divide $360$ by $1$ .

$360$

Step 3.2.7

The period of the $\sin (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 90 + 360 n$ , for any integer $n$

Step 4

Set $\sin (θ) + 4$ equal to $0$ and solve for $θ$ .

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Step 4.1

Set $\sin (θ) + 4$ equal to $0$ .

$\sin (θ) + 4 = 0$

Step 4.2

Solve $\sin (θ) + 4 = 0$ for $θ$ .

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$\sin (θ) = - 4$

Step 4.2.2

The range of sine is $- 1 \leq y \leq 1$ . Since $- 4$ does not fall in this range, there is no solution.

No solution

Step 5

The final solution is all the values that make $(\sin (θ) - 1) (\sin (θ) + 4) = 0$ true.

$θ = 90 + 360 n$ , for any integer $n$