Which Equation Is Not An Identity?A. Tan ⁡ ( − T ) = − Tan ⁡ T \tan (-t) = -\tan T Tan ( − T ) = − Tan T B. Tan ⁡ T ( Csc ⁡ T + Cot ⁡ T ) = Sec ⁡ T + 1 \tan T (\csc T + \cot T) = \sec T + 1 Tan T ( Csc T + Cot T ) = Sec T + 1 C. Sin ⁡ T Cot ⁡ T = Cos ⁡ T \sin T \cot T = \cos T Sin T Cot T = Cos T D. Cos ⁡ 2 T − Sin ⁡ 2 T = − 1 \cos^2 T - \sin^2 T = -1 Cos 2 T − Sin 2 T = − 1

Introduction

In mathematics, an identity is an equation that is true for all possible values of the variables involved. In this article, we will examine four trigonometric equations and determine which one is not an identity. We will use the definitions of the trigonometric functions and various trigonometric identities to analyze each equation.

Equation A: $\tan (-t) = -\tan t$

The tangent function is defined as the ratio of the sine and cosine functions:

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

Using the definition of the tangent function, we can rewrite Equation A as:

$$\frac{\sin (-t)}{\cos (-t)} = -\frac{\sin t}{\cos t}$$

Since the sine and cosine functions are odd and even functions, respectively, we have:

$$\sin (-t) = -\sin t$$

$$\cos (-t) = \cos t$$

Substituting these expressions into Equation A, we get:

$$\frac{-\sin t}{\cos t} = -\frac{\sin t}{\cos t}$$

This equation is true for all values of $t$, so Equation A is an identity.

Equation B: $\tan t (\csc t + \cot t) = \sec t + 1$

The cosecant and cotangent functions are defined as the reciprocals of the sine and cosine functions, respectively:

$$\csc t = \frac{1}{\sin t}$$

$$\cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t}$$

Using these definitions, we can rewrite Equation B as:

$$\tan t \left( \frac{1}{\sin t} + \frac{\cos t}{\sin t} \right) = \frac{1}{\cos t} + 1$$

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

$$\frac{\tan t}{\sin t} + \frac{\tan t \cos t}{\sin t} = \frac{1}{\cos t} + 1$$

Using the definition of the tangent function, we can rewrite the left-hand side of the equation as:

$$\frac{\sin t}{\cos t} + \frac{\cos t \sin t}{\cos t} = \frac{1}{\cos t} + 1$$

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

$$\frac{\sin t + \cos^2 t}{\cos t} = \frac{1}{\cos t} + 1$$

Multiplying both sides of the equation by $\cos t$, we get:

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

Subtracting $\cos^2 t$ from both sides of the equation, we get:

$$\sin t = 1$$

This equation is not true for all values of $t$, so Equation B is not an identity.

Equation C: $\sin t \cot t = \cos t$

The cotangent function is defined as the reciprocal of the tangent function:

$$\cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t}$$

Using this definition, we can rewrite Equation C as:

$$\sin t \frac{\cos t}{\sin t} = \cos t$$

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

$$\cos t = \cos t$$

This equation is true for all values of $t$, so Equation C is an identity.

Equation D: $\cos^2 t - \sin^2 t = -1$

Using the Pythagorean identity, we can rewrite Equation D as:

$$\cos^2 t - \sin^2 t = \cos^2 t + \sin^2 t$$

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

$$\cos^2 t - \sin^2 t = 1$$

This equation is not true for all values of $t$, so Equation D is not an identity.

Conclusion

In this article, we examined four trigonometric equations and determined which one is not an identity. We found that Equation B is not an identity, while Equations A, C, and D are identities. We used the definitions of the trigonometric functions and various trigonometric identities to analyze each equation.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Trigonometric Identities" by Paul Dawkins

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry

Frequently Asked Questions

• Q: What is an identity in mathematics? A: An identity is an equation that is true for all possible values of the variables involved.
• Q: How do you determine if an equation is an identity? A: You can use the definitions of the trigonometric functions and various trigonometric identities to analyze the equation.
• Q: What are some common trigonometric identities? A: Some common trigonometric identities include the Pythagorean identity, the sum and difference formulas, and the double-angle and half-angle formulas.
  Frequently Asked Questions: 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 involves the use of trigonometric functions such as sine, cosine, and tangent to solve problems involving right triangles.

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 Pythagorean identity?

A: The Pythagorean identity is a fundamental trigonometric identity that states:

sin^2(x) + cos^2(x) = 1

This identity is used to relate the sine and cosine functions to each other.

Q: What are the sum and difference formulas?

A: The sum and difference formulas are used to find the sine, cosine, and tangent of the sum or difference of two angles. They are:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b) sin(a - b) = sin(a)cos(b) - cos(a)sin(b) cos(a + b) = cos(a)cos(b) - sin(a)sin(b) cos(a - b) = cos(a)cos(b) + sin(a)sin(b) tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)) tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

Q: What are the double-angle and half-angle formulas?

A: The double-angle and half-angle formulas are used to find the sine, cosine, and tangent of twice or half an angle. They are:

sin(2x) = 2sin(x)cos(x) cos(2x) = cos^2(x) - sin^2(x) tan(2x) = 2tan(x) / (1 - tan^2(x)) sin(x/2) = ±√((1 - cos(x)) / 2) cos(x/2) = ±√((1 + cos(x)) / 2) tan(x/2) = ±√((1 - cos(x)) / (1 + cos(x)))

Q: How do I use trigonometry to solve problems?

A: To use trigonometry to solve problems, you need to:

1. Identify the type of problem you are trying to solve (e.g. finding the length of a side, the measure of an angle, etc.)
2. Choose the appropriate trigonometric function to use (e.g. sine, cosine, tangent, etc.)
3. Use the trigonometric identity or formula to relate the given information to the unknown quantity
4. Solve for the unknown quantity

Q: What are some common applications of trigonometry?

A: Trigonometry has many applications in various fields, including:

• Navigation: trigonometry is used to calculate distances and directions between two points
• Physics: trigonometry is used to describe the motion of objects and the forces acting on them
• Engineering: trigonometry is used to design and build structures such as bridges and buildings
• Computer graphics: trigonometry is used to create 3D models and animations

Q: How can I practice trigonometry?

A: You can practice trigonometry by:

• Working through practice problems and exercises
• Using online resources and calculators to check your work
• Joining a study group or finding a study partner to work through problems together
• Taking online courses or tutorials to learn more about trigonometry

Q: What are some common mistakes to avoid when working with trigonometry?

A: Some common mistakes to avoid when working with trigonometry include:

• Not using the correct trigonometric function for the problem
• Not simplifying expressions correctly
• Not using the correct trigonometric identity or formula
• Not checking your work for errors

Q: How can I use trigonometry to solve real-world problems?

A: You can use trigonometry to solve real-world problems by:

• Identifying the type of problem you are trying to solve (e.g. finding the length of a side, the measure of an angle, etc.)
• Choosing the appropriate trigonometric function to use (e.g. sine, cosine, tangent, etc.)
• Using the trigonometric identity or formula to relate the given information to the unknown quantity
• Solving for the unknown quantity

Q: What are some common real-world applications of trigonometry?

A: Some common real-world applications of trigonometry include:

• Navigation: trigonometry is used to calculate distances and directions between two points
• Physics: trigonometry is used to describe the motion of objects and the forces acting on them
• Engineering: trigonometry is used to design and build structures such as bridges and buildings
• Computer graphics: trigonometry is used to create 3D models and animations

Q: How can I use trigonometry to improve my problem-solving skills?

A: You can use trigonometry to improve your problem-solving skills by:

• Practicing regularly to build your skills and confidence
• Working through practice problems and exercises
• Using online resources and calculators to check your work
• Joining a study group or finding a study partner to work through problems together
• Taking online courses or tutorials to learn more about trigonometry