Consider the given information.
sin θ =1/4; 0^(∘)< θ <90^(∘)
We want to find the exact values of sin 2θ, cos 2θ, sin θ2, and cos θ2.
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.
Value of sin 2θ
To find the value of sin 2θ, we will use one Pythagorean Identity and one Double-Angle Identity.
Pythagorean Identity
Double-Angle Identity
cos ^2 θ + sin ^2 θ =1
sin 2θ=2sin θ cos θ
Let's start by finding the value of cos θ using the Pythagorean Identity in the above table.
We found that cos θ = ± sqrt(15)4. Now we need to find the sign of cos θ. Recall that we were told that θ is greater than 0^(∘) and less than 90^(∘). This means that if we draw the angle θ in standard position, the terminal side will be located in Quadrant I.
We can see that cos θ is positive for θ greater than 0^(∘) and less than 90^(∘).
cos θ = sqrt(15)/4
Finally, we will substitute this value and the value of sin θ into our Double-Angle Identity.
To find the value of cos 2θ, we will use only one Double-Angle Identity.
cos 2θ = 2cos ^2 θ -1
We previously found that cos θ = sqrt(15)4, so we can substitute this value into the identity and simplify.
To find the value of sin θ2 we will use one Half-Angle Identity.
sin θ/2 = ± sqrt(1-cos θ/2)
We previously found that cos θ = sqrt(15)4 so we can substitute this value in the identity and simplify.
Now let's determine the sign of sin θ2. Knowing that θ is greater than 0^(∘) and less than 90^(∘), we can determine the range of values for θ2.
0^(∘) < θ < 90^(∘) ⇔ 0^(∘) < θ/2 < 45^(∘)
Since θ2 is greater than 0^(∘) and less than 45^(∘), it is located in Quadrant I.
Therefore, sin θ2 is positive.
sin θ/2= sqrt(8-2sqrt(15))/4
Value of cos θ2
To find the value of cos θ2 we will use one Half-Angle Identity.
cos θ/2 = ± sqrt(1+cos θ/2)
We previously found that cos θ = sqrt(15)4. Let's substitute this value into the identity and simplify.
