Simplify:$ \left(\frac{1}{\csc \theta}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta }$Options 1. 12. { \sec^2 \theta$ $3. { \csc^2 \theta$}$

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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 simplifying trigonometric expressions, specifically the given expression (1csc⁡θ)(1sin⁡θ)sin⁡2θ\left(\frac{1}{\csc \theta}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Given Expression

The given expression is (1csc⁡θ)(1sin⁡θ)sin⁡2θ\left(\frac{1}{\csc \theta}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta. To simplify this expression, we need to understand the definitions of the trigonometric functions involved.

  • csc⁡θ\csc \theta is the cosecant of angle θ\theta, which is defined as the reciprocal of the sine of θ\theta. In other words, csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.
  • sin⁡θ\sin \theta is the sine of angle θ\theta, which is a ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

Step 1: Simplify the Expression Using the Definition of Cosecant

We can start by simplifying the expression using the definition of cosecant. Since csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, we can rewrite the given expression as:

(1csc⁡θ)(1sin⁡θ)sin⁡2θ=(11sin⁡θ)(1sin⁡θ)sin⁡2θ\left(\frac{1}{\csc \theta}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta = \left(\frac{1}{\frac{1}{\sin \theta}}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta

Step 2: Simplify the Expression Using the Reciprocal Identity

We can simplify the expression further using the reciprocal identity, which states that 11a=a\frac{1}{\frac{1}{a}} = a. Applying this identity to the expression, we get:

(11sin⁡θ)(1sin⁡θ)sin⁡2θ=sin⁡θ(1sin⁡θ)sin⁡2θ\left(\frac{1}{\frac{1}{\sin \theta}}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta = \sin \theta \left(\frac{1}{\sin \theta}\right) \sin^2 \theta

Step 3: Simplify the Expression Using the Distributive Property

We can simplify the expression further using the distributive property, which states that a(b+c)=ab+aca(b + c) = ab + ac. Applying this property to the expression, we get:

sin⁡θ(1sin⁡θ)sin⁡2θ=sin⁡θ(sin⁡2θsin⁡θ)\sin \theta \left(\frac{1}{\sin \theta}\right) \sin^2 \theta = \sin \theta \left(\frac{\sin^2 \theta}{\sin \theta}\right)

Step 4: Simplify the Expression Using the Canceling Property

We can simplify the expression further using the canceling property, which states that aa=1\frac{a}{a} = 1. Applying this property to the expression, we get:

sin⁡θ(sin⁡2θsin⁡θ)=sin⁡θsin⁡θ\sin \theta \left(\frac{\sin^2 \theta}{\sin \theta}\right) = \sin \theta \sin \theta

Step 5: Simplify the Expression Using the Multiplication Property

We can simplify the expression further using the multiplication property, which states that a⋅a=a2a \cdot a = a^2. Applying this property to the expression, we get:

sin⁡θsin⁡θ=sin⁡2θ\sin \theta \sin \theta = \sin^2 \theta

Conclusion

In conclusion, we have simplified the given expression (1csc⁡θ)(1sin⁡θ)sin⁡2θ\left(\frac{1}{\csc \theta}\right)\left(\frac{1}{\sin \theta}\right) \sin^2 \theta using the definitions of the trigonometric functions involved, the reciprocal identity, the distributive property, the canceling property, and the multiplication property. The simplified expression is sin⁡2θ\sin^2 \theta.

Final Answer

The final answer is sin⁡2θ\boxed{\sin^2 \theta}.

Discussion

The given expression is a classic example of a trigonometric expression that can be simplified using various identities and properties. The solution involves breaking down the expression into manageable steps and applying the relevant identities and properties at each step. The final answer is sin⁡2θ\sin^2 \theta, which is a fundamental trigonometric expression that has numerous applications in various fields.

Related Topics

  • Trigonometric identities
  • Reciprocal identities
  • Distributive property
  • Canceling property
  • Multiplication property

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Trigonometric Identities" by Paul Dawkins
  • [3] "Algebra and Trigonometry" by James Stewart

FAQs

  • Q: What is the definition of cosecant? A: The cosecant of an angle θ\theta is the reciprocal of the sine of θ\theta, i.e., csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.
  • Q: What is the reciprocal identity? A: The reciprocal identity states that 11a=a\frac{1}{\frac{1}{a}} = a.
  • Q: What is the distributive property? A: The distributive property states that a(b+c)=ab+aca(b + c) = ab + ac.
  • Q: What is the canceling property? A: The canceling property states that aa=1\frac{a}{a} = 1.
  • Q: What is the multiplication property? A: The multiplication property states that a⋅a=a2a \cdot a = a^2.

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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&A

Q: What is the definition of sine, cosine, and tangent?

A: The sine, cosine, and tangent of an angle θ\theta are defined as follows:

  • sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
  • cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
  • tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Q: What are the reciprocal identities?

A: The reciprocal identities are:

  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

Q: What is the Pythagorean identity?

A: The Pythagorean identity is:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Q: What is the sum and difference formulas?

A: The sum and difference formulas are:

  • sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A + B) = \sin A \cos B + \cos A \sin B
  • sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B
  • cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B
  • cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos (A - B) = \cos A \cos B + \sin A \sin B

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

A: The double-angle and half-angle formulas are:

  • sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta
  • cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta
  • sin⁡θ2=±1−cos⁡θ2\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}
  • cos⁡θ2=±1+cos⁡θ2\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}

Q: What is the law of sines and the law of cosines?

A: The law of sines and the law of cosines are:

  • sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
  • a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A

Q: What is the unit circle and its significance?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is significant because it helps to define the trigonometric functions and their values for various angles.

Conclusion

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. The questions and answers provided in this article cover some of the most important concepts and formulas in trigonometry. We hope that this article has been helpful in answering your questions and providing a better understanding of trigonometry.

Final Thoughts

Trigonometry is a fascinating subject that has been studied for centuries. It has numerous applications in various fields, including physics, engineering, and navigation. The concepts and formulas covered in this article are just a few of the many that are used in trigonometry. We hope that this article has been helpful in providing a better understanding of trigonometry and its applications.

Related Topics

  • Trigonometric identities
  • Reciprocal identities
  • Pythagorean identity
  • Sum and difference formulas
  • Double-angle and half-angle formulas
  • Law of sines and law of cosines
  • Unit circle

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Trigonometric Identities" by Paul Dawkins
  • [3] "Algebra and Trigonometry" by James Stewart

FAQs

  • Q: What is the definition of sine, cosine, and tangent? A: The sine, cosine, and tangent of an angle θ\theta are defined as follows:
    • sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
    • cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
    • tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
  • Q: What are the reciprocal identities? A: The reciprocal identities are:
    • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}
    • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
    • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}
  • Q: What is the Pythagorean identity? A: The Pythagorean identity is:
    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • Q: What is the sum and difference formulas? A: The sum and difference formulas are:
    • sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A + B) = \sin A \cos B + \cos A \sin B
    • sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B
    • cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B
    • cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos (A - B) = \cos A \cos B + \sin A \sin B
  • Q: What is the double-angle and half-angle formulas? A: The double-angle and half-angle formulas are:
    • sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta
    • cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \theta
    • sin⁡θ2=±1−cos⁡θ2\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}
    • cos⁡θ2=±1+cos⁡θ2\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}
  • Q: What is the law of sines and the law of cosines? A: The law of sines and the law of cosines are:
    • sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
    • a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
  • Q: What is the unit circle and its significance? A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is significant because it helps to define the trigonometric functions and their values for various angles.