Trigonometry Examples

$x \sin^{2} (45) = \tan^{2} (60)$

Step 1

Simplify $x \sin^{2} (45)$ .

Tap for more steps...

Step 1.1

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$x {(\frac{\sqrt{2}}{2})}^{2} = \tan^{2} (60)$

Step 1.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$x \frac{{\sqrt{2}}^{2}}{2^{2}} = \tan^{2} (60)$

Step 1.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} = \tan^{2} (60)$

Step 1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} = \tan^{2} (60)$

Step 1.3.3

Combine $\frac{1}{2}$ and $2$ .

$x \frac{2^{\frac{2}{2}}}{2^{2}} = \tan^{2} (60)$

Step 1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.4.1

Cancel the common factor.

$x \frac{2^{\frac{2}{2}}}{2^{2}} = \tan^{2} (60)$

Step 1.3.4.2

Rewrite the expression.

$x \frac{2^{1}}{2^{2}} = \tan^{2} (60)$

Step 1.3.5

Evaluate the exponent.

$x \frac{2}{2^{2}} = \tan^{2} (60)$

Step 1.4

Raise $2$ to the power of $2$ .

$x \frac{2}{4} = \tan^{2} (60)$

Step 1.5

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 1.5.1

Factor $2$ out of $2$ .

$x \frac{2 (1)}{4} = \tan^{2} (60)$

Step 1.5.2

Cancel the common factors.

Tap for more steps...

Step 1.5.2.1

Factor $2$ out of $4$ .

$x \frac{2 \cdot 1}{2 \cdot 2} = \tan^{2} (60)$

Step 1.5.2.2

Cancel the common factor.

$x \frac{2 \cdot 1}{2 \cdot 2} = \tan^{2} (60)$

Step 1.5.2.3

Rewrite the expression.

$x \frac{1}{2} = \tan^{2} (60)$

Step 1.6

Combine $x$ and $\frac{1}{2}$ .

$\frac{x}{2} = \tan^{2} (60)$

Step 2

Simplify $\tan^{2} (60)$ .

Tap for more steps...

Step 2.1

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{x}{2} = {\sqrt{3}}^{2}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{x}{2} = {(3^{\frac{1}{2}})}^{2}$

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x}{2} = 3^{\frac{1}{2} \cdot 2}$

Step 2.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{x}{2} = 3^{\frac{2}{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

$\frac{x}{2} = 3^{\frac{2}{2}}$

Step 2.2.4.2

Rewrite the expression.

$\frac{x}{2} = 3^{1}$

Step 2.2.5

Evaluate the exponent.

$\frac{x}{2} = 3$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.14146497$

Step 4