Recalling Key Trigonometric Identities

Oct 13, 2025 by miodeya.com 21 views

Understanding the Basics of Trigonometry

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 discussion, we will focus on identifying the false statement among the given options.

Recalling Key Trigonometric Identities

Before we proceed, let's recall some key trigonometric identities that will help us evaluate the given statements.

Tangent and Cotangent: The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The cotangent of an angle is the reciprocal of the tangent, i.e., cot θ = 1/tan θ.
Sine and Cosine: The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Pythagorean Identity: The Pythagorean identity states that sin^2 θ + cos^2 θ = 1 for any angle θ.

Evaluating the Statements

Now, let's evaluate each statement:

A. tan 45^{\circ} = cot 45^{\circ}

We know that tan 45^{\circ} = 1 and cot 45^{\circ} = 1. Therefore, statement A is TRUE.

B. sin 90^{\circ} = tan 45^{\circ}

We know that sin 90^{\circ} = 1 and tan 45^{\circ} = 1. Therefore, statement B is TRUE.

C. sin 30^{\circ} = cos 30^{\circ}

We know that sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2. Therefore, statement C is FALSE.

D. sin 45^{\circ} = cos 45^{\circ}

We know that sin 45^{\circ} = √2/2 and cos 45^{\circ} = √2/2. Therefore, statement D is TRUE.

Conclusion

In conclusion, the false statement among the given options is C. sin 30^{\circ} = cos 30^{\circ}. This statement is false because sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2.

Key Takeaways

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Key trigonometric identities include the definitions of tangent and cotangent, sine and cosine, and the Pythagorean identity.
The false statement among the given options is C. sin 30^{\circ} = cos 30^{\circ}.

Final Thoughts

In this discussion, we evaluated four statements involving trigonometric functions. We recalled key trigonometric identities and used them to determine which statement is false. The false statement is C. sin 30^{\circ} = cos 30^{\circ}. This statement is false because sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2.

Understanding the Basics of Trigonometry

Recalling Key Trigonometric Identities

Before we proceed, let's recall some key trigonometric identities that will help us evaluate the given statements.

Tangent and Cotangent: The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The cotangent of an angle is the reciprocal of the tangent, i.e., cot θ = 1/tan θ.
Sine and Cosine: The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Pythagorean Identity: The Pythagorean identity states that sin^2 θ + cos^2 θ = 1 for any angle θ.

Evaluating the Statements

Now, let's evaluate each statement:

A. tan 45^{\circ} = cot 45^{\circ}

We know that tan 45^{\circ} = 1 and cot 45^{\circ} = 1. Therefore, statement A is TRUE.

B. sin 90^{\circ} = tan 45^{\circ}

We know that sin 90^{\circ} = 1 and tan 45^{\circ} = 1. Therefore, statement B is TRUE.

C. sin 30^{\circ} = cos 30^{\circ}

We know that sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2. Therefore, statement C is FALSE.

D. sin 45^{\circ} = cos 45^{\circ}

We know that sin 45^{\circ} = √2/2 and cos 45^{\circ} = √2/2. Therefore, statement D is TRUE.

Q&A Session

Q: What is the definition of tangent and cotangent?

A: The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The cotangent of an angle is the reciprocal of the tangent, i.e., cot θ = 1/tan θ.

Q: What is the Pythagorean identity?

A: The Pythagorean identity states that sin^2 θ + cos^2 θ = 1 for any angle θ.

Q: Why is statement C false?

A: Statement C is false because sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2.

Q: What is the relationship between sine and cosine?

A: The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Conclusion

In conclusion, the false statement among the given options is C. sin 30^{\circ} = cos 30^{\circ}. This statement is false because sin 30^{\circ} = 1/2 and cos 30^{\circ} = √3/2.

Key Takeaways

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Key trigonometric identities include the definitions of tangent and cotangent, sine and cosine, and the Pythagorean identity.
The false statement among the given options is C. sin 30^{\circ} = cos 30^{\circ}.