Whose Procedure Is Correct?A. Keisha's Procedure Is Correct. B. David's Procedure Is Correct. C. Both Procedures Are Correct. D. Neither Procedure Is Correct. Given:$[ \begin{aligned} \frac{\left(-\frac{8}{17}\right)^2}{\cos ^2 \theta}+1 &

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Introduction

In mathematics, trigonometric functions and their properties are crucial for solving various problems. The given problem involves the trigonometric function cosine and its relationship with a fraction. We are asked to determine whose procedure is correct in simplifying the given expression. In this article, we will analyze the procedures of Keisha and David and determine which one is correct.

The Given Expression

The given expression is:

(−817)2cos⁡2θ+1\frac{\left(-\frac{8}{17}\right)^2}{\cos ^2 \theta}+1

Keisha's Procedure

Keisha's procedure involves squaring the fraction and then dividing it by the cosine squared of theta. She then adds 1 to the result.

Step 1: Square the Fraction

Keisha starts by squaring the fraction:

(−817)2=64289\left(-\frac{8}{17}\right)^2 = \frac{64}{289}

Step 2: Divide by Cosine Squared

Next, Keisha divides the squared fraction by the cosine squared of theta:

64289cos⁡2θ\frac{\frac{64}{289}}{\cos ^2 \theta}

Step 3: Add 1

Finally, Keisha adds 1 to the result:

64289cos⁡2θ+1\frac{\frac{64}{289}}{\cos ^2 \theta} + 1

David's Procedure

David's procedure involves using the trigonometric identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1 to simplify the expression.

Step 1: Use the Trigonometric Identity

David starts by using the trigonometric identity:

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

Step 2: Rearrange the Identity

David rearranges the identity to isolate the cosine squared of theta:

cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta

Step 3: Substitute the Expression

David substitutes the expression for cosine squared of theta into the original expression:

(−817)21−sin⁡2θ+1\frac{\left(-\frac{8}{17}\right)^2}{1 - \sin^2 \theta} + 1

Analysis

Now that we have analyzed the procedures of Keisha and David, let's determine which one is correct.

Keisha's Procedure

Keisha's procedure involves squaring the fraction and then dividing it by the cosine squared of theta. However, this procedure does not take into account the trigonometric identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1. Therefore, Keisha's procedure is not correct.

David's Procedure

David's procedure involves using the trigonometric identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1 to simplify the expression. This procedure takes into account the relationship between the cosine squared of theta and the sine squared of theta. Therefore, David's procedure is correct.

Conclusion

In conclusion, David's procedure is correct. He used the trigonometric identity cos⁡2θ+sin⁡2θ=1\cos^2 \theta + \sin^2 \theta = 1 to simplify the expression, which is a more accurate and efficient way to solve the problem.

Final Answer

The final answer is:

B. David's procedure is correct.

Discussion

The given problem involves the trigonometric function cosine and its relationship with a fraction. We analyzed the procedures of Keisha and David and determined that David's procedure is correct. This problem requires a good understanding of trigonometric functions and their properties.

Trigonometric Functions

Trigonometric functions are used to describe the relationships between the sides and angles of triangles. The six basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are used to solve problems involving right triangles, circular functions, and trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables. These identities are used to simplify expressions and solve problems involving trigonometric functions. Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Applications of Trigonometry

Trigonometry has many applications in various fields, including physics, engineering, navigation, and computer science. Some common applications of trigonometry include:

  • Calculating distances and heights
  • Determining the angles of triangles
  • Solving problems involving circular functions
  • Modeling periodic phenomena

Conclusion

In conclusion, trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Trigonometric functions and their properties are used to solve problems involving right triangles, circular functions, and trigonometric identities. The given problem requires a good understanding of trigonometric functions and their properties.

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many applications in various fields, including physics, engineering, navigation, and computer science. In this article, we will answer some common questions about trigonometry.

Q1: What is the difference between sine, cosine, and tangent?

A1: Sine, cosine, and tangent are three basic trigonometric functions. Sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Cosine is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Tangent is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q2: What is the Pythagorean identity?

A2: The Pythagorean identity is a fundamental trigonometric identity that states: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. This identity is used to simplify expressions and solve problems involving trigonometric functions.

Q3: How do I use the trigonometric identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}?

A3: To use the trigonometric identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, you need to know the values of sine and cosine of the angle. You can then substitute these values into the identity to find the value of tangent.

Q4: What is the difference between a right triangle and an oblique triangle?

A4: A right triangle is a triangle with one right angle (90 degrees). An oblique triangle is a triangle with no right angles. Trigonometry is used to solve problems involving both right and oblique triangles.

Q5: How do I use the law of sines to solve a problem?

A5: The law of sines is a fundamental trigonometric identity that states: sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}. To use the law of sines, you need to know the values of the angles and the lengths of the sides of the triangle. You can then substitute these values into the identity to solve the problem.

Q6: What is the difference between a circular function and a trigonometric function?

A6: A circular function is a function that is periodic and has a period of 2π2\pi. A trigonometric function is a function that is used to describe the relationships between the sides and angles of triangles. Circular functions are used to model periodic phenomena, while trigonometric functions are used to solve problems involving triangles.

Q7: How do I use the trigonometric identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}?

A7: To use the trigonometric identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, you need to know the value of cosine of the angle. You can then substitute this value into the identity to find the value of secant.

Q8: What is the difference between a trigonometric identity and a trigonometric equation?

A8: A trigonometric identity is an equation that is true for all values of the variables. A trigonometric equation is an equation that is true for a specific value of the variable. Trigonometric identities are used to simplify expressions and solve problems involving trigonometric functions, while trigonometric equations are used to solve problems involving specific values of the variables.

Q9: How do I use the trigonometric identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}?

A9: To use the trigonometric identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, you need to know the value of sine of the angle. You can then substitute this value into the identity to find the value of cosecant.

Q10: What is the difference between a trigonometric function and a trigonometric ratio?

A10: A trigonometric function is a function that is used to describe the relationships between the sides and angles of triangles. A trigonometric ratio is a ratio of the lengths of the sides of a triangle. Trigonometric functions are used to solve problems involving triangles, while trigonometric ratios are used to describe the relationships between the sides and angles of triangles.

Conclusion

In conclusion, trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many applications in various fields, including physics, engineering, navigation, and computer science. The questions and answers in this article provide a brief overview of the basics of trigonometry and its applications.

Final Answer

The final answer is:

  • Q1: Sine, cosine, and tangent are three basic trigonometric functions.
  • Q2: The Pythagorean identity is a fundamental trigonometric identity that states: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Q3: To use the trigonometric identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, you need to know the values of sine and cosine of the angle.
  • Q4: A right triangle is a triangle with one right angle (90 degrees), while an oblique triangle is a triangle with no right angles.
  • Q5: The law of sines is a fundamental trigonometric identity that states: sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.
  • Q6: A circular function is a function that is periodic and has a period of 2π2\pi, while a trigonometric function is a function that is used to describe the relationships between the sides and angles of triangles.
  • Q7: To use the trigonometric identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, you need to know the value of cosine of the angle.
  • Q8: A trigonometric identity is an equation that is true for all values of the variables, while a trigonometric equation is an equation that is true for a specific value of the variable.
  • Q9: To use the trigonometric identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, you need to know the value of sine of the angle.
  • Q10: A trigonometric function is a function that is used to describe the relationships between the sides and angles of triangles, while a trigonometric ratio is a ratio of the lengths of the sides of a triangle.