1.2 If Sin ⁡ 17 ∘ = A \sin 17^{\circ}=a Sin 1 7 ∘ = A , WITHOUT Using A Calculator, Express The Following In Terms Of A A A :1.2.1 $\tan 17^{\circ}$1.2.2 $\sin 107^{\circ}$1.2.3 $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$1.3 Simplify

Introduction

In this article, we will explore the process of expressing various trigonometric functions in terms of a single angle, specifically $\sin 17^{\circ}=a$. We will use the given information to derive expressions for $\tan 17^{\circ}$, $\sin 107^{\circ}$, and $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$ without relying on a calculator.

Expressing $\tan 17^{\circ}$ in Terms of $a$

To express $\tan 17^{\circ}$ in terms of $a$, we can use the definition of the tangent function:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Since we are given that $\sin 17^{\circ}=a$, we can rewrite the tangent function as:

$$\tan 17^{\circ} = \frac{a}{\cos 17^{\circ}}$$

To find the value of $\cos 17^{\circ}$, we can use the Pythagorean identity:

$$\sin^2 \theta + \cos^2 \theta = 1$$

Substituting $\sin 17^{\circ}=a$, we get:

$$a^2 + \cos^2 17^{\circ} = 1$$

Solving for $\cos^2 17^{\circ}$, we get:

$$\cos^2 17^{\circ} = 1 - a^2$$

Now, we can substitute this expression into the tangent function:

$$\tan 17^{\circ} = \frac{a}{\sqrt{1 - a^2}}$$

Expressing $\sin 107^{\circ}$ in Terms of $a$

To express $\sin 107^{\circ}$ in terms of $a$, we can use the fact that $\sin (180^{\circ} - \theta) = \sin \theta$:

$$\sin 107^{\circ} = \sin (180^{\circ} - 73^{\circ}) = \sin 73^{\circ}$$

Using the angle subtraction formula for sine, we get:

$$\sin 73^{\circ} = \sin (90^{\circ} - 17^{\circ}) = \cos 17^{\circ}$$

Since we are given that $\sin 17^{\circ}=a$, we can rewrite the expression as:

$$\sin 107^{\circ} = \cos 17^{\circ} = \sqrt{1 - a^2}$$

Expressing $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$ in Terms of $a$

To express $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$ in terms of $a$, we can use the fact that $\cos (180^{\circ} + \theta) = -\cos \theta$ and $\sin (180^{\circ} + \theta) = -\sin \theta$:

$$\cos ^2 253^{\circ}+\sin ^2 557^{\circ} = \cos ^2 (180^{\circ} + 73^{\circ}) + \sin ^2 (180^{\circ} + 73^{\circ})$$

$$= (-\cos 73^{\circ})^2 + (-\sin 73^{\circ})^2$$

$$= \cos ^2 73^{\circ} + \sin ^2 73^{\circ}$$

Using the Pythagorean identity, we get:

$$\cos ^2 73^{\circ} + \sin ^2 73^{\circ} = 1$$

However, we can also express $\cos 73^{\circ}$ and $\sin 73^{\circ}$ in terms of $a$ using the angle addition formula:

$$\cos 73^{\circ} = \cos (90^{\circ} - 17^{\circ}) = \sin 17^{\circ} = a$$

$$\sin 73^{\circ} = \sin (90^{\circ} - 17^{\circ}) = \cos 17^{\circ} = \sqrt{1 - a^2}$$

Substituting these expressions into the original equation, we get:

$$\cos ^2 253^{\circ}+\sin ^2 557^{\circ} = a^2 + (1 - a^2)$$

$$= 1$$

Conclusion

In this article, we have expressed various trigonometric functions in terms of a single angle, specifically $\sin 17^{\circ}=a$. We have derived expressions for $\tan 17^{\circ}$, $\sin 107^{\circ}$, and $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$ without relying on a calculator. These expressions can be used to simplify trigonometric expressions and solve problems involving right triangles.

References

• [1] "Trigonometry" by Michael Corral, 2018.
• [2] "A Survey of Mathematics with Applications" by Allen R. Angel, 2013.

Glossary

• Tangent: The ratio of the sine of an angle to the cosine of the same angle.
• Pythagorean identity: The equation $\sin^2 \theta + \cos^2 \theta = 1$.
• Angle addition formula: The equation $\sin (A + B) = \sin A \cos B + \cos A \sin B$.
• Angle subtraction formula: The equation $\sin (A - B) = \sin A \cos B - \cos A \sin B$.
  Q&A: Expressing Trigonometric Functions in Terms of a Single Angle ====================================================================

Introduction

In our previous article, we explored the process of expressing various trigonometric functions in terms of a single angle, specifically $\sin 17^{\circ}=a$. We derived expressions for $\tan 17^{\circ}$, $\sin 107^{\circ}$, and $\cos ^2 253^{\circ}+\sin ^2 557^{\circ}$ without relying on a calculator. In this article, we will answer some common questions related to expressing trigonometric functions in terms of a single angle.

Q: What is the significance of expressing trigonometric functions in terms of a single angle?

A: Expressing trigonometric functions in terms of a single angle is important because it allows us to simplify complex trigonometric expressions and solve problems involving right triangles. By expressing a trigonometric function in terms of a single angle, we can use the properties of that angle to simplify the expression and make it easier to work with.

Q: How do I express $\tan \theta$ in terms of $\sin \theta$ and $\cos \theta$?

A: To express $\tan \theta$ in terms of $\sin \theta$ and $\cos \theta$, you can use the definition of the tangent function:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Q: How do I express $\sin (A + B)$ in terms of $\sin A$ and $\sin B$?

A: To express $\sin (A + B)$ in terms of $\sin A$ and $\sin B$, you can use the angle addition formula:

$$\sin (A + B) = \sin A \cos B + \cos A \sin B$$

Q: How do I express $\cos (A - B)$ in terms of $\cos A$ and $\cos B$?

A: To express $\cos (A - B)$ in terms of $\cos A$ and $\cos B$, you can use the angle subtraction formula:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Q: What is the Pythagorean identity, and how is it used in trigonometry?

A: The Pythagorean identity is the equation $\sin^2 \theta + \cos^2 \theta = 1$. It is used in trigonometry to relate the sine and cosine of an angle to each other. By using the Pythagorean identity, we can express one trigonometric function in terms of another.

Q: How do I use the Pythagorean identity to express $\cos \theta$ in terms of $\sin \theta$?

A: To use the Pythagorean identity to express $\cos \theta$ in terms of $\sin \theta$, you can rearrange the equation to solve for $\cos \theta$:

$$\cos \theta = \pm \sqrt{1 - \sin^2 \theta}$$

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that relate the ratio of the opposite side to the hypotenuse in a right triangle. However, the sine function is defined as the ratio of the opposite side to the hypotenuse, while the cosine function is defined as the ratio of the adjacent side to the hypotenuse.

Q: How do I use the sine and cosine functions to solve problems involving right triangles?

A: To use the sine and cosine functions to solve problems involving right triangles, you can use the definitions of the functions to relate the sides of the triangle to each other. By using the sine and cosine functions, you can solve for the lengths of the sides of the triangle and find the angles.

Conclusion

In this article, we have answered some common questions related to expressing trigonometric functions in terms of a single angle. We have discussed the significance of expressing trigonometric functions in terms of a single angle, and we have provided examples of how to use the Pythagorean identity and the angle addition and subtraction formulas to express trigonometric functions in terms of a single angle. By understanding how to express trigonometric functions in terms of a single angle, you can simplify complex trigonometric expressions and solve problems involving right triangles.

References

• [1] "Trigonometry" by Michael Corral, 2018.
• [2] "A Survey of Mathematics with Applications" by Allen R. Angel, 2013.

Glossary

• Tangent: The ratio of the sine of an angle to the cosine of the same angle.
• Pythagorean identity: The equation $\sin^2 \theta + \cos^2 \theta = 1$.
• Angle addition formula: The equation $\sin (A + B) = \sin A \cos B + \cos A \sin B$.
• Angle subtraction formula: The equation $\sin (A - B) = \sin A \cos B - \cos A \sin B$.
• Sine: The ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine: The ratio of the adjacent side to the hypotenuse in a right triangle.