Establish The Identity.$\sin \theta(\cot \theta+\tan \theta)=\sec \theta$Write The Left Side In Terms Of Sine And Cosine:$\sin \theta(\square$\]

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Establish the Identity: sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta

In trigonometry, identities are essential to simplify complex expressions and solve problems. One such identity is sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta. In this article, we will explore this identity, break it down, and rewrite the left side in terms of sine and cosine.

Before we dive into the solution, let's understand the components of the identity. We have three trigonometric functions involved:

  • sin⁡θ\sin \theta: Sine of an angle
  • cot⁡θ\cot \theta: Cotangent of an angle (defined as cos⁡θsin⁡θ\frac{\cos \theta}{\sin \theta})
  • tan⁡θ\tan \theta: Tangent of an angle (defined as sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta})
  • sec⁡θ\sec \theta: Secant of an angle (defined as 1cos⁡θ\frac{1}{\cos \theta})

The identity states that the product of sin⁡θ\sin \theta and the sum of cot⁡θ\cot \theta and tan⁡θ\tan \theta is equal to sec⁡θ\sec \theta.

To rewrite the left side of the identity in terms of sine and cosine, we need to simplify the expression sin⁡θ(cot⁡θ+tan⁡θ)\sin \theta(\cot \theta+\tan \theta). Let's start by expanding the product:

sin⁡θ(cot⁡θ+tan⁡θ)=sin⁡θ⋅cot⁡θ+sin⁡θ⋅tan⁡θ\sin \theta(\cot \theta+\tan \theta) = \sin \theta \cdot \cot \theta + \sin \theta \cdot \tan \theta

Now, let's substitute the definitions of cot⁡θ\cot \theta and tan⁡θ\tan \theta:

sin⁡θ⋅cos⁡θsin⁡θ+sin⁡θ⋅sin⁡θcos⁡θ\sin \theta \cdot \frac{\cos \theta}{\sin \theta} + \sin \theta \cdot \frac{\sin \theta}{\cos \theta}

Simplifying the expressions, we get:

cos⁡θ+sin⁡2θ⋅1cos⁡θ\cos \theta + \sin^2 \theta \cdot \frac{1}{\cos \theta}

Now, let's rewrite the left side of the identity in terms of sine and cosine. We can start by simplifying the expression cos⁡θ+sin⁡2θ⋅1cos⁡θ\cos \theta + \sin^2 \theta \cdot \frac{1}{\cos \theta}:

cos⁡θ+sin⁡2θcos⁡θ\cos \theta + \frac{\sin^2 \theta}{\cos \theta}

Using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we can rewrite the expression as:

cos⁡θ+1−cos⁡2θcos⁡θ\cos \theta + \frac{1 - \cos^2 \theta}{\cos \theta}

Simplifying further, we get:

cos⁡θ+1cos⁡θ−cos⁡θ\cos \theta + \frac{1}{\cos \theta} - \cos \theta

Cancelling out the cos⁡θ\cos \theta terms, we are left with:

1cos⁡θ\frac{1}{\cos \theta}

In this article, we established the identity sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta and rewrote the left side in terms of sine and cosine. We started by breaking down the identity, expanding the product, and substituting the definitions of cot⁡θ\cot \theta and tan⁡θ\tan \theta. We then simplified the expressions and used the Pythagorean identity to rewrite the left side. Finally, we arrived at the conclusion that the left side of the identity is equal to 1cos⁡θ\frac{1}{\cos \theta}, which is equivalent to sec⁡θ\sec \theta. This identity is a useful tool for simplifying complex trigonometric expressions and solving problems in mathematics and physics.

The final answer is 1cos⁡θ\boxed{\frac{1}{\cos \theta}}.
Q&A: Establishing the Identity sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta

In our previous article, we established the identity sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta and rewrote the left side in terms of sine and cosine. In this article, we will answer some frequently asked questions related to this identity.

A: This identity is significant because it provides a relationship between the sine and cosine functions, which are fundamental trigonometric functions. It can be used to simplify complex expressions and solve problems in mathematics and physics.

A: This identity is used in various real-world applications, such as:

  • Physics: In the study of waves and vibrations, this identity is used to describe the behavior of waves and vibrations in terms of sine and cosine functions.
  • Engineering: In the design of electronic circuits and systems, this identity is used to analyze and optimize circuit performance.
  • Computer Science: In the development of algorithms and data structures, this identity is used to optimize computational complexity and improve performance.

A: Here are a few examples:

  • Example 1: Simplify the expression sin⁡θ(cot⁡θ+tan⁡θ)\sin \theta(\cot \theta+\tan \theta) using the identity.
    • Solution: Using the identity, we can rewrite the expression as sec⁡θ\sec \theta.
  • Example 2: Solve the equation sin⁡θ(cot⁡θ+tan⁡θ)=2\sin \theta(\cot \theta+\tan \theta) = 2 using the identity.
    • Solution: Using the identity, we can rewrite the equation as sec⁡θ=2\sec \theta = 2. Then, we can solve for θ\theta using the inverse secant function.
  • Example 3: Simplify the expression cos⁡θ+sin⁡2θ⋅1cos⁡θ\cos \theta + \sin^2 \theta \cdot \frac{1}{\cos \theta} using the identity.
    • Solution: Using the identity, we can rewrite the expression as sec⁡θ\sec \theta.

A: Here are a few common mistakes to avoid:

  • Mistake 1: Not recognizing the identity and trying to simplify the expression using other methods.
  • Mistake 2: Not using the correct trigonometric functions and identities.
  • Mistake 3: Not checking the domain and range of the functions and identities used.

A: Here are a few additional resources:

  • Textbooks: "Trigonometry" by Michael Corral, "Calculus" by Michael Spivak
  • Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
  • Videos: 3Blue1Brown, Crash Course, Khan Academy

In this article, we answered some frequently asked questions related to the identity sin⁡θ(cot⁡θ+tan⁡θ)=sec⁡θ\sin \theta(\cot \theta+\tan \theta)=\sec \theta. We provided examples of how to use this identity in problems and discussed common mistakes to avoid. We also provided additional resources for learning more about this identity. We hope this article has been helpful in establishing a deeper understanding of this identity and its applications.