4.2.2 Calculate The Value Of $\tan 15^{\circ}$.4.3 Given That $\sqrt{2} \sin \left(x+45^{\circ}\right)=\sin X+\cos X$:4.3.1 Prove This Identity.4.3.2 Deduce The Minimum Value Of \$\sin X+\cos X$[/tex\] And Give The

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically the calculation of the value of $\tan 15^{\circ}$ and the proof of the identity $\sqrt{2} \sin \left(x+45^{\circ}\right)=\sin x+\cos x$.

Calculating the Value of $\tan 15^{\circ}$

To calculate the value of $\tan 15^{\circ}$, we can use the half-angle formula for tangent:

$$\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta}$$

In this case, we have $\theta = 30^{\circ}$, so we can substitute this value into the formula:

$$\tan 15^{\circ} = \tan \frac{30^{\circ}}{2} = \frac{1 - \cos 30^{\circ}}{\sin 30^{\circ}}$$

We know that $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$, so we can substitute these values into the formula:

$$\tan 15^{\circ} = \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3}$$

Therefore, the value of $\tan 15^{\circ}$ is $2 - \sqrt{3}$.

Proving the Identity $\sqrt{2} \sin \left(x+45^{\circ}\right)=\sin x+\cos x$

To prove this identity, we can start by using the angle addition formula for sine:

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

In this case, we have $A = x$ and $B = 45^{\circ}$, so we can substitute these values into the formula:

$$\sin (x + 45^{\circ}) = \sin x \cos 45^{\circ} + \cos x \sin 45^{\circ}$$

We know that $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$ and $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$, so we can substitute these values into the formula:

$$\sin (x + 45^{\circ}) = \sin x \frac{\sqrt{2}}{2} + \cos x \frac{\sqrt{2}}{2}$$

Multiplying both sides of the equation by $\sqrt{2}$, we get:

$$\sqrt{2} \sin (x + 45^{\circ}) = \sqrt{2} \sin x \frac{\sqrt{2}}{2} + \sqrt{2} \cos x \frac{\sqrt{2}}{2}$$

Simplifying the right-hand side of the equation, we get:

$$\sqrt{2} \sin (x + 45^{\circ}) = \sin x + \cos x$$

Therefore, we have proved the identity $\sqrt{2} \sin \left(x+45^{\circ}\right)=\sin x+\cos x$.

Deducing the Minimum Value of $\sin x + \cos x$

To deduce the minimum value of $\sin x + \cos x$, we can use the fact that the minimum value of a sine function is $-1$ and the minimum value of a cosine function is $-1$. Therefore, the minimum value of $\sin x + \cos x$ is $-1 + (-1) = -2$.

However, we can also use the identity $\sin x + \cos x = \sqrt{2} \sin (x + 45^{\circ})$ to deduce the minimum value of $\sin x + \cos x$. Since the minimum value of a sine function is $-1$, the minimum value of $\sqrt{2} \sin (x + 45^{\circ})$ is $-\sqrt{2}$. Therefore, the minimum value of $\sin x + \cos x$ is $-\sqrt{2}$.

Conclusion

In this article, we have solved the trigonometric equation $\tan 15^{\circ}$ and proved the identity $\sqrt{2} \sin \left(x+45^{\circ}\right)=\sin x+\cos x$. We have also deduced the minimum value of $\sin x + \cos x$, which is $-\sqrt{2}$. These results demonstrate the importance of trigonometry in solving mathematical problems and have numerous applications in various fields.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Engineers" by John Bird

Further Reading

• [1] "Trigonometry for Dummies" by Mary Jane Sterling
• [2] "Calculus for Dummies" by Mark Ryan
• [3] "Mathematics for Engineers: A Comprehensive Guide" by John Bird

Glossary

• Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Tangent: A trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.
• Sine: A trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse.
• Cosine: A trigonometric function that is defined as the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.
• Angle addition formula: A formula that is used to find the sine of the sum of two angles.
• Minimum value: The smallest possible value of a function.
  Trigonometry Q&A: Frequently Asked Questions =====================================================

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some of the most frequently asked questions about trigonometry.

Q: What is trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

• Sine (sin): The ratio of the length of the side opposite a given angle to the length of the hypotenuse.
• Cosine (cos): The ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.
• Tangent (tan): The ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.

Q: What is the difference between sine and cosine?

A: The sine and cosine functions are both trigonometric functions that are used to describe the relationships between the sides and angles of triangles. However, the sine function is used to describe the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while the cosine function is used to describe the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle. It is often used to describe the steepness of a line or the angle of a triangle.

Q: How do I use trigonometry in real life?

A: Trigonometry has numerous applications in various fields, including physics, engineering, and navigation. Some examples of how trigonometry is used in real life include:

• Building design: Trigonometry is used to design buildings and ensure that they are structurally sound.
• Physics: Trigonometry is used to describe the motion of objects and the forces that act upon them.
• Navigation: Trigonometry is used to determine the location of objects and navigate through space.
• Computer graphics: Trigonometry is used to create 3D models and animations.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Angle addition formula: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• Angle subtraction formula: $\sin (A - B) = \sin A \cos B - \cos A \sin B$

Q: How do I solve trigonometric equations?

A: To solve trigonometric equations, you can use a variety of techniques, including:

• Using trigonometric identities: Trigonometric identities can be used to simplify and solve trigonometric equations.
• Using algebraic techniques: Algebraic techniques, such as factoring and solving quadratic equations, can be used to solve trigonometric equations.
• Using numerical methods: Numerical methods, such as the Newton-Raphson method, can be used to solve trigonometric equations.

Conclusion

In this article, we have answered some of the most frequently asked questions about trigonometry. We hope that this article has been helpful in providing a better understanding of trigonometry and its applications.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Engineers" by John Bird

Further Reading

• [1] "Trigonometry for Dummies" by Mary Jane Sterling
• [2] "Calculus for Dummies" by Mark Ryan
• [3] "Mathematics for Engineers: A Comprehensive Guide" by John Bird

Glossary

• Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
• Tangent: A trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.
• Sine: A trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse.
• Cosine: A trigonometric function that is defined as the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.
• Angle addition formula: A formula that is used to find the sine of the sum of two angles.
• Minimum value: The smallest possible value of a function.