Question

Determine the exact values of the other five trigonometric ratios under the given conditions. a) \(\sin \theta=\frac{3}{5}, \frac{\pi}{2}<\theta<\pi\) b) \(\cos \theta=\frac{-2 \sqrt{2}}{3},-\pi \leq \theta \leq \frac{3 \pi}{2}\) c) \(\tan \theta=\frac{2}{3},-360^{\circ}<\theta<180^{\circ}\) d) \(\sec \theta=\frac{4 \sqrt{3}}{3},-180^{\circ} \leq \theta \leq 180^{\circ}\)

Short answer

\text{cos} \theta = -\frac{4}{5}, \text{tan} \theta = -\frac{3}{4}, \text{csc} \theta = \frac{5}{3}, \text{sec} \theta = -\frac{5}{4}, \text{cot} \theta = -\frac{4}{3}.

Step-by-step solution

Determine the quadrant and sign

Use \(\frac{π}{2} < \theta < π\) to determine \(\theta\) is in Quadrant II, where sine is positive and cosine, tangent, cotangent, secant, and cosecant are negative.

Compute cosine using Pythagorean identity

Using \(\text{sin} \theta = \frac{3}{5}\), find cosine: \[ \text{cos}^2 \theta + \text{sin}^2 \theta = 1 \] \[ \text{cos}^2 \theta = 1 - \text{sin}^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \] \[ \text{cos} \theta = -\frac{4}{5} \]

Compute tangent using sine and cosine

Using \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \): \[ \text{tan} \theta = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} \]

Compute cosecant, secant, and cotangent

Using reciprocal identities: \(\text{sin} \theta = \frac{3}{5} \), find \(\text{csc} \theta = \frac{1}{\text{sin} \theta} = \frac{5}{3} \). \[ \text{sec} \theta = \frac{1}{\text{cos} \theta} = -\frac{5}{4} \] \[ \text{cot} \theta = \frac{1}{\text{tan} \theta} = -\frac{4}{3} \]

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities help simplify complex trigonometric expressions and solve various equations. The most common trigonometric identities include:

• Pythagorean identities
• Reciprocal identities
• Quotient identities
• Co-function identities
• Even-Odd identities

For example, given the trigonometric identity for sine and cosine in the context of a triangle, we can write: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). This Pythagorean identity is derived from the Pythagorean theorem, applied to a right-angled triangle.

Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry. It states that for any angle \( \theta \): \[ \text{cos}^2 \theta + \text{sin}^2 \theta = 1 \]To see how this is used, let’s look at the given problem: \( \text{sin} \theta = \frac{3}{5} \). Using the Pythagorean identity, we find \( \text{cos} \theta \): \[\text{cos}^2 \theta = 1 - \text{sin}^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}\]Since \( \frac{\text{π}}{2} < \theta < \text{π} \) (indicating that \( \theta \) is in the second quadrant), we know cosine is negative. Thus, \( \text{cos} \theta = -\frac{4}{5} \). This identity is crucial because it allows us to find one trigonometric ratio knowing another.

Reciprocal Identities

Reciprocal identities relate the six trigonometric functions to each other as reciprocals:

• \( \text{sin} \theta \) and \( \text{csc} \theta \)
• \( \text{cos} \theta \) and \( \text{sec} \theta \)
• \( \text{tan} \theta \) and \( \text{cot} \theta \)

For example, if we know \( \text{sin} \theta \), we can find \( \text{csc} \theta \) as its reciprocal: \[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]Given \( \text{sin} \theta = \frac{3}{5} \), we find: \[ \text{csc} \theta = \frac{5}{3} \]Similarly, for \( \text{cos} \theta = -\frac{4}{5} \), its reciprocal is \( \text{sec} \theta \): \[ \text{sec} \theta = \frac{1}{\text{cos} \theta} = -\frac{5}{4} \]Using these identities simplifies finding all trigonometric ratios once any one of them is known.

Quadrant Determination

Determining the quadrant of an angle helps understand the sign and value of trigonometric functions. The coordinate plane is divided into four quadrants:

• Quadrant I: All functions are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In the given example, \( \frac{π}{2} < \theta < π \) indicates \( \theta \) is in Quadrant II. Here, sine is positive, while cosine and tangent are negative. This knowledge helps us correctly assign the signs to the calculated trigonometric values, ensuring accurate answers for problems involving trigonometric functions.