Consider the given information.

sin θ = \frac{1}{4}; 0^{\circ} < θ < 9 0^{\circ}

We want to find the exact values of

sin 2 θ,

cos 2 θ,

sin \frac{θ}{2},

and

cos \frac{θ}{2} .

To do so, we will use some of the Pythagorean Identities, Double-Angle Identities, and Half-Angle Identities. Let's find the desired values one at a time.

To find the value of $sin 2 θ,$ we will use one Pythagorean Identity and one Double-Angle Identity.

Pythagorean Identity	Double-Angle Identity
$cos^{2} θ + sin^{2} θ = 1$	$sin 2 θ = 2 sin θ cos θ$

Let's start by finding the value of

cos θ

using the Pythagorean Identity in the above table.

cos^{2} θ + sin^{2} θ = 1

$sin θ = \frac{1}{4}$

cos^{2} θ + (\frac{1}{4})^{2} = 1

▼

Calculate power

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

cos^{2} θ + \frac{1 ^{2}}{4 ^{2}} = 1

Calculate power

cos^{2} θ + \frac{1}{1 6} = 1

▼

Solve for

cos θ

$LHS - \frac{1}{1 6} = RHS - \frac{1}{1 6}$

cos^{2} θ = 1 - \frac{1}{1 6}

Rewrite $1$ as $\frac{1 6}{1 6}$

cos^{2} θ = \frac{1 6}{1 6} - \frac{1}{1 6}

Subtract fractions

cos^{2} θ = \frac{1 5}{1 6}

$LHS = RHS$

cos θ = \pm \frac{1 5}{1 6}

$\frac{a}{b} = \frac{a}{b}$

cos θ = \pm \frac{1 5}{1 6}

Calculate root

cos θ = \pm \frac{1 5}{4}

We found that

cos θ = \pm \frac{1 5}{4} .

Now we need to find the sign of

cos θ .

Recall that we were told that

θ

is greater than

0^{\circ}

and less than

9 0^{\circ} .

This means that if we draw the angle

θ

in standard position, the terminal side will be located in Quadrant I.

Exercise