Question

If \(\cos \theta=\frac{5}{7}\) and \(\tan \theta<0,\) what is the value of \(\csc \theta ?\)

Short answer

- \frac{7\sqrt{6}}{12}

Step-by-step solution

- Identify the quadrant

Since \(\tan \theta < 0\) and \(\tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\), it means that sine and cosine have different signs. Cosine is positive, so sine must be negative. Thus, \(\theta\) must be in the fourth quadrant.

- Use the Pythagorean identity

Recall the Pythagorean identity: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1.\) Substitute the given value of \(\text{cos} \theta = \frac{5}{7}\) into the identity: \[ \text{sin}^2 \theta + \left(\frac{5}{7}\right)^2 = 1. \]

- Solve for \(\text{sin}^2 \theta\)

Simplify the equation: \[ \text{sin}^2 \theta + \frac{25}{49} = 1. \] \[\text{sin}^2\theta = 1 - \frac{25}{49} = \frac{49}{49} - \frac{25}{49} = \frac{24}{49}.\]

- Determine \(\text{sin} \theta\)

Since \(\text{sin} \theta\) must be negative in the fourth quadrant, take the negative square root: \[\text{sin} \theta = -\sqrt{\frac{24}{49}} = -\frac{\sqrt{24}}{7} = -\frac{2\sqrt{6}}{7}.\]

- Calculate \(\text{csc} \theta\)

Recall that \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). Substitute the value of \(\text{sin} \theta\): \[\text{csc} \theta = \frac{1}{-\frac{2\sqrt{6}}{7}} = -\frac{7}{2\sqrt{6}}.\] Rationalize the denominator to get: \[ \text{csc }\theta = -\frac{7\sqrt{6}}{12}. \]

Key concepts

Pythagorean Identity

One of the most fundamental identities in trigonometry is the Pythagorean identity. It is given by \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). This relationship tells us that the square of the sine of an angle plus the square of the cosine of the same angle always equals one.
To understand why this identity is so useful, let's look at how it helps when solving problems. For example, if we know \(\text{cos} \theta\) and need to find \(\text{sin} \theta\), we can simply rearrange the Pythagorean identity to \(\text{sin}^2 \theta = 1 - \text{cos}^2 \theta\).
This powerful identity stems from the geometry of a right triangle inscribed within a unit circle. It can be applied in various trigonometric problems to find missing values and to simplify expressions.

Quadrant Identification

Identifying the right quadrant is crucial in trigonometry, especially since trigonometric functions have different signs in different quadrants. Angles in the unit circle are typically measured starting from the positive x-axis and moving counterclockwise.
The four quadrants are:

• First Quadrant (0° to 90°): All trigonometric functions are positive.
• Second Quadrant (90° to 180°): Sine is positive, but cosine and tangent are negative.
• Third Quadrant (180° to 270°): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (270° to 360°): Cosine is positive, but sine and tangent are negative.

In our exercise, since \(\text{cos} \theta\) is positive and \(\text{tan} \theta < 0\), we determine that \(\theta\) is in the fourth quadrant. This helps us figure out that \(\text{sin} \theta\) must be negative.

Cosecant Calculation

The cosecant function, denoted as \(\text{csc} \theta\), is the reciprocal of the sine function: \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). This means that to find \(\text{csc} \theta\), we first need to find \(\text{sin} \theta\).
In the given problem, we found \(\text{sin} \theta\) to be \(-\frac{2\frac{6}}{7}\). Therefore, \(\text{csc} \theta = \frac{1}{-\frac{2\frac{6}}{7}}\).
Next, we simplify this by flipping the fraction, giving us \(\text{csc} \theta = -\frac{7}{2\frac{6}}\).

Rationalizing Denominators

Rationalizing the denominator is a key step in simplifying fractions that contain a radical (square root) in the denominator. The aim is to eliminate the radical from the denominator by multiplying the numerator and the denominator by a suitable term.
In our exercise, we had \(\text{csc} \theta = -\frac{7}{2\frac{6}}\). To rationalize this, multiply both top and bottom by \(\frac{6}\):

• Numerator: \(-7\frac{6}\)
• Denominator: \(2\frac{6}\times\frac{6} = 12\)

So, \(\text{csc} \theta\) simplifies to \(-\frac{7\frac{6}}{12}\). Rationalizing denominators ensures that we express our answers in a standardized form, making them easier to read and understand.