Solving Trigonometric Equations Check All That Apply If Csc Θ=13/5

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In the realm of trigonometry, understanding the relationships between trigonometric functions is crucial for solving various problems. This article delves into a specific problem where $\csc \theta = \frac{13}{5}$, and we aim to determine which of the given options are correct. We will explore the fundamental trigonometric identities and apply them step-by-step to arrive at the solutions. This comprehensive guide is designed to provide a clear and thorough understanding of the concepts involved, making it easier for students and enthusiasts to tackle similar problems.

Decoding the Trigonometric Puzzle: If $\csc \theta = \frac{13}{5}$, What's the Next Step?

When faced with a trigonometric problem like this, the first step is to identify the given information and what we need to find. Here, we are given that the cosecant of angle $\theta$ (csctheta{csc \\theta}) is $\frac{13}{5}$. Our task is to evaluate the following statements and determine which ones hold true:

  • A. $\tan \theta = \frac{5}{12}$
  • B. $\sin \theta = \frac{5}{13}$
  • C. $\cos \theta = \frac{5}{13}$
  • D. $\sec \theta = \frac{5}{13}$

To solve this, we need to understand the relationships between the trigonometric functions, particularly the reciprocal identities and the Pythagorean identity. Let's start by revisiting these fundamental concepts.

The Foundation: Reciprocal Identities

The reciprocal identities are the cornerstone of solving this problem. These identities define the relationships between sine, cosine, tangent, and their reciprocals—cosecant, secant, and cotangent. Specifically:

  • csctheta=frac1sintheta\\csc \\theta = \\frac{1}{\\sin \\theta}

  • sectheta=frac1costheta\\sec \\theta = \\frac{1}{\\cos \\theta}

  • cottheta=frac1tantheta\\cot \\theta = \\frac{1}{\\tan \\theta}

Given that $\csc \theta = \frac{13}{5}$, we can immediately find $\sin \theta$ using the reciprocal identity. This will help us verify option B.

Unveiling Sine: Finding $\sin \theta$

Using the reciprocal identity, we know that $\sin \theta$ is the reciprocal of $\csc \theta$. Therefore:

sintheta=frac1csctheta=frac1frac135=frac513\\sin \\theta = \\frac{1}{\\csc \\theta} = \\frac{1}{\\frac{13}{5}} = \\frac{5}{13}

This confirms that option B, $\sin \theta = \frac{5}{13}$, is correct. Now, let's move on to the next step, which involves using the Pythagorean identity to find $\cos \theta$.

The Pythagorean Identity: Linking Sine and Cosine

The Pythagorean identity is another essential tool in trigonometry. It relates sine and cosine through the equation:

sin2theta+cos2theta=1\\sin^2 \\theta + \\cos^2 \\theta = 1

We already know $\sin \theta = \frac{5}{13}$, so we can substitute this value into the Pythagorean identity and solve for $\cos \theta$.

Calculating Cosine: Solving for $\cos \theta$

Substituting $\sin \theta = \frac{5}{13}$ into the Pythagorean identity, we get:

left(frac513right)2+cos2theta=1\\left(\\frac{5}{13}\\right)^2 + \\cos^2 \\theta = 1

frac25169+cos2theta=1\\\\frac{25}{169} + \\cos^2 \\theta = 1

Now, we solve for $\cos^2 \theta$:

cos2theta=1−frac25169=frac169−25169=frac144169\\cos^2 \\theta = 1 - \\frac{25}{169} = \\frac{169 - 25}{169} = \\frac{144}{169}

Taking the square root of both sides, we find:

costheta=pmsqrtfrac144169=pmfrac1213\\cos \\theta = \\pm\\sqrt{\\frac{144}{169}} = \\pm\\frac{12}{13}

Since we are not given the quadrant of $\theta$, we will consider both positive and negative values for $\cos \theta$. However, the options provided do not include a negative value for cosine, so we will primarily focus on the positive value, $\cos \theta = \frac{12}{13}$. This result indicates that option C, $\cos \theta = \frac{5}{13}$, is incorrect.

Finding Secant: The Reciprocal of Cosine

Now that we have found $\cos \theta$, we can determine $\sec \theta$ using the reciprocal identity:

sectheta=frac1costheta\\sec \\theta = \\frac{1}{\\cos \\theta}

Using the positive value of $\cos \theta = \frac{12}{13}$, we get:

sectheta=frac1frac1213=frac1312\\sec \\theta = \\frac{1}{\\frac{12}{13}} = \\frac{13}{12}

This shows that option D, $\sec \theta = \frac{5}{13}$, is also incorrect. The correct value for $\sec \theta$ is $\frac{13}{12}$.

Determining Tangent: Sine Divided by Cosine

To find $\tan \theta$, we use the identity:

tantheta=fracsinthetacostheta\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}

We know that $\sin \theta = \frac{5}{13}$ and $\cos \theta = \pm\frac{12}{13}$. Substituting these values, we get:

tantheta=fracfrac513pmfrac1213=pmfrac512\\tan \\theta = \\frac{\\frac{5}{13}}{\\pm\\frac{12}{13}} = \\pm\\frac{5}{12}

Considering the positive value, $\tan \theta = \frac{5}{12}$, we find that option A, $\tan \theta = \frac{5}{12}$, is correct. If we consider the negative value, $\tan \theta = -\frac{5}{12}$, this would also be a valid solution depending on the quadrant of $\theta$. However, since only the positive value is provided in the options, we will focus on that.

Conclusion: Summarizing the Correct Options

In summary, given that $\csc \theta = \frac{13}{5}$, we have determined the following:

  • \\sin \\theta = \\frac{5}{13}$ (Option B is correct)

  • costheta=pmfrac1213\\cos \\theta = \\pm\\frac{12}{13}

  • sectheta=frac13pm12\\sec \\theta = \\frac{13}{\\pm 12}

  • \\tan \\theta = \\pm\\frac{5}{12}$ (Option A is correct)

Therefore, the correct options are A and B. This step-by-step solution illustrates the importance of understanding and applying trigonometric identities to solve problems effectively. By using the reciprocal and Pythagorean identities, we were able to find the values of sine, cosine, secant, and tangent based on the given cosecant value.

Trigonometry, the branch of mathematics that deals with the relationships between the sides and angles of triangles, is a fundamental topic in mathematics. It has numerous applications in fields such as engineering, physics, and computer graphics. One of the core concepts in trigonometry is the understanding of trigonometric functions, their properties, and their interrelationships. This article aims to provide a detailed explanation of how to solve a trigonometric problem where the cosecant of an angle (csctheta{csc \\theta}) is given, and we need to determine the values of other trigonometric functions. Specifically, we will address the question: If $\csc \theta = \frac{13}{5}$, which of the following statements are true?

  • A. $\tan \theta = \frac{5}{12}$
  • B. $\sin \theta = \frac{5}{13}$
  • C. $\cos \theta = \frac{5}{13}$
  • D. $\sec \theta = \frac{5}{13}$

This problem requires a solid understanding of the basic trigonometric identities and how they relate to each other. We will begin by revisiting the definitions of the trigonometric functions and their reciprocals. Then, we will use these definitions and the Pythagorean identity to solve the problem step by step. This detailed approach will not only help in solving this specific problem but also in building a strong foundation for tackling more complex trigonometric questions.

Understanding Trigonometric Functions and Their Reciprocals

To effectively solve the given problem, it is crucial to have a clear understanding of the basic trigonometric functions and their reciprocals. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which are defined in terms of the ratios of the sides of a right-angled triangle. Their reciprocals are cosecant (csc), secant (sec), and cotangent (cot), respectively. Let's define these functions more formally:

  • Sine (sin): The sine of an angle $\theta$ in a right-angled triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.
  • Cosine (cos): The cosine of an angle $\theta$ is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Mathematically, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
  • Tangent (tan): The tangent of an angle $\theta$ is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.

The reciprocal functions are defined as follows:

  • Cosecant (csc): The cosecant of an angle $\theta$ is the reciprocal of the sine function. Mathematically, $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$.
  • Secant (sec): The secant of an angle $\theta$ is the reciprocal of the cosine function. Mathematically, $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$.
  • Cotangent (cot): The cotangent of an angle $\theta$ is the reciprocal of the tangent function. Mathematically, $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}$.

Understanding these definitions is the first step in solving our problem. We are given that $\csc \theta = \frac{13}{5}$, which means the ratio of the hypotenuse to the opposite side is $\frac{13}{5}$. From this, we can deduce the value of $\sin \theta$ immediately.

Determining $\sin \theta$ Using the Reciprocal Identity

Since $\csc \theta$ is the reciprocal of $\sin \theta$, we can easily find $\sin \theta$ by taking the reciprocal of the given value:

sintheta=frac1csctheta\\sin \\theta = \\frac{1}{\\csc \\theta}

Given that $\csc \theta = \frac{13}{5}$, we have:

sintheta=frac1frac135=frac513\\sin \\theta = \\frac{1}{\\frac{13}{5}} = \\frac{5}{13}

This confirms that option B, which states $\sin \theta = \frac{5}{13}$, is correct. Now, we need to determine whether the other options are correct. To do this, we will use the Pythagorean identity, which relates sine and cosine, to find the value of $\cos \theta$.

Applying the Pythagorean Identity to Find $\cos \theta$

The Pythagorean identity is a fundamental trigonometric identity that states:

sin2theta+cos2theta=1\\sin^2 \\theta + \\cos^2 \\theta = 1

This identity is derived from the Pythagorean theorem applied to a right-angled triangle. We know the value of $\sin \theta$, so we can substitute it into the Pythagorean identity and solve for $\cos \theta$:

left(frac513right)2+cos2theta=1\\left(\\frac{5}{13}\\right)^2 + \\cos^2 \\theta = 1

frac25169+cos2theta=1\\\\frac{25}{169} + \\cos^2 \\theta = 1

Now, we isolate $\cos^2 \theta$:

cos2theta=1−frac25169\\cos^2 \\theta = 1 - \\frac{25}{169}

cos2theta=frac169−25169\\cos^2 \\theta = \\frac{169 - 25}{169}

cos2theta=frac144169\\cos^2 \\theta = \\frac{144}{169}

Taking the square root of both sides, we get:

costheta=pmsqrtfrac144169=pmfrac1213\\cos \\theta = \\pm\\sqrt{\\frac{144}{169}} = \\pm\\frac{12}{13}

So, $\cos \theta$ can be either $\frac{12}{13}$ or $\-\frac{12}{13}$. Option C states that $\cos \theta = \frac{5}{13}$, which is incorrect. Therefore, option C is not a valid answer.

Calculating $\sec \theta$ and Checking Option D

Next, we need to find the value of $\sec \theta$ to check option D. Secant is the reciprocal of cosine, so:

sectheta=frac1costheta\\sec \\theta = \\frac{1}{\\cos \\theta}

Using the positive value of $\cos \theta = \frac{12}{13}$, we get:

sectheta=frac1frac1213=frac1312\\sec \\theta = \\frac{1}{\\frac{12}{13}} = \\frac{13}{12}

Using the negative value of $\cos \theta = \-\frac{12}{13}$, we get:

sectheta=frac1−frac1213=−frac1312\\sec \\theta = \\frac{1}{\\-\\frac{12}{13}} = -\\frac{13}{12}

Option D states that $\sec \theta = \frac{5}{13}$, which is incorrect. Therefore, option D is also not a valid answer. The correct value for $\sec \theta$ is $\pm\frac{13}{12}$.

Determining $\tan \theta$ and Evaluating Option A

Finally, we need to find the value of $\tan \theta$ to evaluate option A. Tangent is defined as the ratio of sine to cosine:

tantheta=fracsinthetacostheta\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}

Using the value of $\sin \theta = \frac{5}{13}$ and the positive value of $\cos \theta = \frac{12}{13}$, we get:

tantheta=fracfrac513frac1213=frac512\\tan \\theta = \\frac{\\frac{5}{13}}{\\frac{12}{13}} = \\frac{5}{12}

Using the value of $\sin \theta = \frac{5}{13}$ and the negative value of $\cos \theta = -\frac{12}{13}$, we get:

tantheta=fracfrac513−frac1213=−frac512\\tan \\theta = \\frac{\\frac{5}{13}}{-\\frac{12}{13}} = -\\frac{5}{12}

Option A states that $\tan \theta = \frac{5}{12}$, which is correct. Therefore, option A is a valid answer.

Conclusion: Identifying the Correct Options

In conclusion, given that $\csc \theta = \frac{13}{5}$, we have determined the following:

  • Option A: $\tan \theta = \frac{5}{12}$ is correct.
  • Option B: $\sin \theta = \frac{5}{13}$ is correct.
  • Option C: $\cos \theta = \frac{5}{13}$ is incorrect.
  • Option D: $\sec \theta = \frac{5}{13}$ is incorrect.

Therefore, the correct options are A and B. This detailed explanation demonstrates how to use trigonometric identities and definitions to solve problems involving trigonometric functions. By understanding the relationships between the functions and applying the Pythagorean identity, we can accurately determine the values of various trigonometric functions given the value of one function.