Find The Exact Value Of $\sin 15^{\circ}$ Without A Calculator.$\sin 15^{\circ} = \frac{\sqrt{[?] - \sqrt{\square}}}{}$Double-Angle Formulas:$\[ \begin{array}{l} \sin (2 \theta) = 2 \sin \theta \cos \theta \\ \cos (2 \theta) =

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Introduction

In trigonometry, finding the exact value of a trigonometric function without a calculator can be a challenging task. One of the most common methods used to find the exact value of a trigonometric function is by using the double-angle formulas. In this article, we will use the double-angle formulas to find the exact value of $\sin 15^{\circ}$ without a calculator.

Double-Angle Formulas

The double-angle formulas are a set of formulas that relate the trigonometric functions of an angle to the trigonometric functions of twice the angle. The two most commonly used double-angle formulas are:

$$\sin (2 \theta) = 2 \sin \theta \cos \theta$$

$$\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta$$

Using the Double-Angle Formulas to Find $\sin 15^{\circ}$

To find the exact value of $\sin 15^{\circ}$, we can use the double-angle formula for sine:

$$\sin (2 \theta) = 2 \sin \theta \cos \theta$$

We know that $15^{\circ} = 2 \times 7.5^{\circ}$, so we can substitute $\theta = 7.5^{\circ}$ into the formula:

$$\sin (2 \times 7.5^{\circ}) = 2 \sin 7.5^{\circ} \cos 7.5^{\circ}$$

$$\sin 15^{\circ} = 2 \sin 7.5^{\circ} \cos 7.5^{\circ}$$

Finding the Values of $\sin 7.5^{\circ}$ and $\cos 7.5^{\circ}$

To find the values of $\sin 7.5^{\circ}$ and $\cos 7.5^{\circ}$, we can use the double-angle formulas again. We know that $7.5^{\circ} = 2 \times 3.75^{\circ}$, so we can substitute $\theta = 3.75^{\circ}$ into the formulas:

$$\sin (2 \times 3.75^{\circ}) = 2 \sin 3.75^{\circ} \cos 3.75^{\circ}$$

$$\sin 7.5^{\circ} = 2 \sin 3.75^{\circ} \cos 3.75^{\circ}$$

$$\cos (2 \times 3.75^{\circ}) = \cos^2 3.75^{\circ} - \sin^2 3.75^{\circ}$$

$$\cos 7.5^{\circ} = \cos^2 3.75^{\circ} - \sin^2 3.75^{\circ}$$

Finding the Values of $\sin 3.75^{\circ}$ and $\cos 3.75^{\circ}$

To find the values of $\sin 3.75^{\circ}$ and $\cos 3.75^{\circ}$, we can use the double-angle formulas again. We know that $3.75^{\circ} = 2 \times 1.875^{\circ}$, so we can substitute $\theta = 1.875^{\circ}$ into the formulas:

$$\sin (2 \times 1.875^{\circ}) = 2 \sin 1.875^{\circ} \cos 1.875^{\circ}$$

$$\sin 3.75^{\circ} = 2 \sin 1.875^{\circ} \cos 1.875^{\circ}$$

$$\cos (2 \times 1.875^{\circ}) = \cos^2 1.875^{\circ} - \sin^2 1.875^{\circ}$$

$$\cos 3.75^{\circ} = \cos^2 1.875^{\circ} - \sin^2 1.875^{\circ}$$

Finding the Values of $\sin 1.875^{\circ}$ and $\cos 1.875^{\circ}$

To find the values of $\sin 1.875^{\circ}$ and $\cos 1.875^{\circ}$, we can use the double-angle formulas again. We know that $1.875^{\circ} = 2 \times 0.9375^{\circ}$, so we can substitute $\theta = 0.9375^{\circ}$ into the formulas:

$$\sin (2 \times 0.9375^{\circ}) = 2 \sin 0.9375^{\circ} \cos 0.9375^{\circ}$$

$$\sin 1.875^{\circ} = 2 \sin 0.9375^{\circ} \cos 0.9375^{\circ}$$

$$\cos (2 \times 0.9375^{\circ}) = \cos^2 0.9375^{\circ} - \sin^2 0.9375^{\circ}$$

$$\cos 1.875^{\circ} = \cos^2 0.9375^{\circ} - \sin^2 0.9375^{\circ}$$

Finding the Values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$

To find the values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$, we can use the fact that $\sin 0^{\circ} = 0$ and $\cos 0^{\circ} = 1$. We can also use the fact that $\sin (90^{\circ} - \theta) = \cos \theta$ and $\cos (90^{\circ} - \theta) = \sin \theta$.

We know that $0.9375^{\circ} = 90^{\circ} - 89.0625^{\circ}$, so we can substitute $\theta = 89.0625^{\circ}$ into the formulas:

$$\sin (90^{\circ} - 89.0625^{\circ}) = \cos 89.0625^{\circ}$$

$$\sin 0.9375^{\circ} = \cos 89.0625^{\circ}$$

$$\cos (90^{\circ} - 89.0625^{\circ}) = \sin 89.0625^{\circ}$$

$$\cos 0.9375^{\circ} = \sin 89.0625^{\circ}$$

Finding the Value of $\sin 15^{\circ}$

Now that we have found the values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$, we can substitute them into the formula for $\sin 1.875^{\circ}$:

$$\sin 1.875^{\circ} = 2 \sin 0.9375^{\circ} \cos 0.9375^{\circ}$$

$$\sin 1.875^{\circ} = 2 \cos 89.0625^{\circ} \sin 89.0625^{\circ}$$

$$\sin 1.875^{\circ} = 2 \cos 89.0625^{\circ} \cos (90^{\circ} - 89.0625^{\circ})$$

$$\sin 1.875^{\circ} = 2 \cos 89.0625^{\circ} \cos 89.0625^{\circ}$$

$$\sin 1.875^{\circ} = 2 \cos^2 89.0625^{\circ}$$

Now that we have found the value of $\sin 1.875^{\circ}$, we can substitute it into the formula for $\sin 3.75^{\circ}$:

$$\sin 3.75^{\circ} = 2 \sin 1.875^{\circ} \cos 1.875^{\circ}$$

$$\sin 3.75^{\circ} = 2 (2 \cos^2 89.0625^{\circ}) \cos 1.875^{\circ}$$

$$\sin 3.75^{\circ} = 4 \cos^2 89.0625^{\circ} \cos 1.875^{\circ}$$

Now that we have found the value of $\sin 3.75^{\circ}$, we can substitute it into the formula for $\sin 7.5^{\circ}$:

$$\sin 7.5^{\circ} = 2 \sin 3.75^{\circ} \cos 3.75^{\circ}$$

$$\sin 7.5^{\circ} = 2 (4 \cos^2 89.0625^{\circ} \cos 1.875^{\circ}) \cos 3.75^{\circ}$$

\sin 7.5^{\circ} = 8 \<br/> **Q&A: Finding the Exact Value of $\sin 15^{\circ}$ without a Calculator** ==================================================================== **Q: What is the double-angle formula for sine?** -------------------------------------------- A: The double-angle formula for sine is: $\sin (2 \theta) = 2 \sin \theta \cos \theta

Q: How can we use the double-angle formula to find the exact value of $\sin 15^{\circ}$?

A: We can use the double-angle formula to find the exact value of $\sin 15^{\circ}$ by substituting $\theta = 7.5^{\circ}$ into the formula:

$$\sin (2 \times 7.5^{\circ}) = 2 \sin 7.5^{\circ} \cos 7.5^{\circ}$$

$$\sin 15^{\circ} = 2 \sin 7.5^{\circ} \cos 7.5^{\circ}$$

Q: How can we find the values of $\sin 7.5^{\circ}$ and $\cos 7.5^{\circ}$?

A: We can find the values of $\sin 7.5^{\circ}$ and $\cos 7.5^{\circ}$ by using the double-angle formulas again. We know that $7.5^{\circ} = 2 \times 3.75^{\circ}$, so we can substitute $\theta = 3.75^{\circ}$ into the formulas:

$$\sin (2 \times 3.75^{\circ}) = 2 \sin 3.75^{\circ} \cos 3.75^{\circ}$$

$$\sin 7.5^{\circ} = 2 \sin 3.75^{\circ} \cos 3.75^{\circ}$$

$$\cos (2 \times 3.75^{\circ}) = \cos^2 3.75^{\circ} - \sin^2 3.75^{\circ}$$

$$\cos 7.5^{\circ} = \cos^2 3.75^{\circ} - \sin^2 3.75^{\circ}$$

Q: How can we find the values of $\sin 3.75^{\circ}$ and $\cos 3.75^{\circ}$?

A: We can find the values of $\sin 3.75^{\circ}$ and $\cos 3.75^{\circ}$ by using the double-angle formulas again. We know that $3.75^{\circ} = 2 \times 1.875^{\circ}$, so we can substitute $\theta = 1.875^{\circ}$ into the formulas:

$$\sin (2 \times 1.875^{\circ}) = 2 \sin 1.875^{\circ} \cos 1.875^{\circ}$$

$$\sin 3.75^{\circ} = 2 \sin 1.875^{\circ} \cos 1.875^{\circ}$$

$$\cos (2 \times 1.875^{\circ}) = \cos^2 1.875^{\circ} - \sin^2 1.875^{\circ}$$

$$\cos 3.75^{\circ} = \cos^2 1.875^{\circ} - \sin^2 1.875^{\circ}$$

Q: How can we find the values of $\sin 1.875^{\circ}$ and $\cos 1.875^{\circ}$?

A: We can find the values of $\sin 1.875^{\circ}$ and $\cos 1.875^{\circ}$ by using the double-angle formulas again. We know that $1.875^{\circ} = 2 \times 0.9375^{\circ}$, so we can substitute $\theta = 0.9375^{\circ}$ into the formulas:

$$\sin (2 \times 0.9375^{\circ}) = 2 \sin 0.9375^{\circ} \cos 0.9375^{\circ}$$

$$\sin 1.875^{\circ} = 2 \sin 0.9375^{\circ} \cos 0.9375^{\circ}$$

$$\cos (2 \times 0.9375^{\circ}) = \cos^2 0.9375^{\circ} - \sin^2 0.9375^{\circ}$$

$$\cos 1.875^{\circ} = \cos^2 0.9375^{\circ} - \sin^2 0.9375^{\circ}$$

Q: How can we find the values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$?

A: We can find the values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$ by using the fact that $\sin 0^{\circ} = 0$ and $\cos 0^{\circ} = 1$. We can also use the fact that $\sin (90^{\circ} - \theta) = \cos \theta$ and $\cos (90^{\circ} - \theta) = \sin \theta$.

We know that $0.9375^{\circ} = 90^{\circ} - 89.0625^{\circ}$, so we can substitute $\theta = 89.0625^{\circ}$ into the formulas:

$$\sin (90^{\circ} - 89.0625^{\circ}) = \cos 89.0625^{\circ}$$

$$\sin 0.9375^{\circ} = \cos 89.0625^{\circ}$$

$$\cos (90^{\circ} - 89.0625^{\circ}) = \sin 89.0625^{\circ}$$

$$\cos 0.9375^{\circ} = \sin 89.0625^{\circ}$$

Q: What is the final value of $\sin 15^{\circ}$?

A: After substituting the values of $\sin 0.9375^{\circ}$ and $\cos 0.9375^{\circ}$ into the formula for $\sin 1.875^{\circ}$, we get:

$$\sin 1.875^{\circ} = 2 \cos^2 89.0625^{\circ}$$

Substituting this value into the formula for $\sin 3.75^{\circ}$, we get:

$$\sin 3.75^{\circ} = 4 \cos^2 89.0625^{\circ} \cos 1.875^{\circ}$$

Substituting this value into the formula for $\sin 7.5^{\circ}$, we get:

$$\sin 7.5^{\circ} = 8 \cos^2 89.0625^{\circ} \cos 3.75^{\circ}$$

Substituting this value into the formula for $\sin 15^{\circ}$, we get:

$$\sin 15^{\circ} = 2 \sin 7.5^{\circ} \cos 7.5^{\circ}$$

$$\sin 15^{\circ} = 2 (8 \cos^2 89.0625^{\circ} \cos 3.75^{\circ}) \cos 7.5^{\circ}$$

$$\sin 15^{\circ} = 16 \cos^2 89.0625^{\circ} \cos 3.75^{\circ} \cos 7.5^{\circ}$$

$$\sin 15^{\circ} = \frac{\sqrt{6 - \sqrt{3}}}{4}$$

Therefore, the final value of $\sin 15^{\circ}$ is $\frac{\sqrt{6 - \sqrt{3}}}{4}$.