Trigonometry Examples

\frac{\cos (θ)}{\sin (θ)} + \frac{\sin (θ)}{\cos (θ)}

$\frac{\cos (θ)}{\sin (θ)} + \frac{\sin (θ)}{\cos (θ)}$

Step 1

Given the expression

a \sin (θ) + b \cos (θ)

$a \sin (θ) + b \cos (θ)$ , find the values of

k

$k$ and

θ

$θ$ .

k = \sqrt{a^{2} + b^{2}}

$k = \sqrt{a^{2} + b^{2}}$

θ = \tan^{-1} (\frac{b}{a})

$θ = \tan^{-1} (\frac{b}{a})$

Step 2

Calculate the value for

k

$k$ by substituting the coefficients from

\frac{\cos (θ)}{\sin (θ)}

$\frac{\cos (θ)}{\sin (θ)}$ and

\frac{\sin (θ)}{\cos (θ)}

$\frac{\sin (θ)}{\cos (θ)}$ into

k = \sqrt{a^{2} + b^{2}}

$k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

One to any power is one.

k = \sqrt{1 + {(1)}^{2}}

$k = \sqrt{1 + {(1)}^{2}}$

Step 2.2

One to any power is one.

k = \sqrt{1 + 1}

$k = \sqrt{1 + 1}$

Step 2.3

Add

1

$1$ and

1

$1$ .

k = \sqrt{2}

$k = \sqrt{2}$

k = \sqrt{2}

$k = \sqrt{2}$

Step 3

Find the value for

θ

$θ$ by substituting the coefficients from

\frac{\cos (θ)}{\sin (θ)}

$\frac{\cos (θ)}{\sin (θ)}$ and

\frac{\sin (θ)}{\cos (θ)}

$\frac{\sin (θ)}{\cos (θ)}$ into

θ = \tan^{-1} (\frac{b}{a})

$θ = \tan^{-1} (\frac{b}{a})$ .

θ = \tan^{-1} (\frac{1}{1})

$θ = \tan^{-1} (\frac{1}{1})$

Step 4

Divide

1

$1$ by

1

$1$ .

\tan^{-1} (1)

$\tan^{-1} (1)$

Step 5

Linear combinations of trigonometric functions dictate that

a \sin (θ) + b \cos (θ) = k \sin (θ + θ)

$a \sin (θ) + b \cos (θ) = k \sin (θ + θ)$ . Substitute the values of

k

$k$ and

θ

$θ$ .

\sqrt{2} \sin (θ + \frac{π}{4})

$\sqrt{2} \sin (θ + \frac{π}{4})$