Problem 51 - Algebra and Trigonometry

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \cos \theta=-\frac{2}{7}, \quad \tan \theta<0 $$

Step-by-Step Solution

Answer

\( \cos\theta = -\frac{2}{7}, \sin\theta = \frac{3\sqrt{5}}{7}, \tan\theta = -\frac{3\sqrt{5}}{2} \).

Step 1: Determine the Quadrant

Since cosine is negative and tangent is negative, \( \theta \) must be in the second quadrant. In the second quadrant, \( \cos(\theta) < 0 \), \( \sin(\theta) > 0 \), and \( \tan(\theta) < 0 \).

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that \( \sin^2(\theta) + \cos^2(\theta) = 1 \). We know that \( \cos(\theta) = -\frac{2}{7} \). Substitute in the identity:\[sin^2(\theta) + \left(-\frac{2}{7}\right)^2 = 1 \]Simplifying gives:\[sin^2(\theta) + \frac{4}{49} = 1 \]\[sin^2(\theta) = 1 - \frac{4}{49} = \frac{45}{49}\]

Step 3: Solve for \( \sin(\theta) \)

Since we are in the second quadrant, \( \sin(\theta) > 0 \). Therefore:\[sin(\theta) = \sqrt{\frac{45}{49}} = \frac{\sqrt{45}}{7} = \frac{3\sqrt{5}}{7}\]

Step 4: Calculate \( \tan(\theta) \)

Using the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \):\[tan(\theta) = \frac{\frac{3\sqrt{5}}{7}}{-\frac{2}{7}} = -\frac{3\sqrt{5}}{2}\]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is an essential building block in the world of trigonometry. It states that for any angle \( \theta \), the square of the sine of an angle plus the square of the cosine of that angle equals 1:\[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This relation emerges from the Pythagorean theorem in a right-angled triangle, where the hypotenuse is normalized to be of unit length. This handy identity allows us to solve problems where one trigonometric function is known, and the other needs to be found. For example, if you know \( \cos(\theta) \), you can easily calculate \( \sin(\theta) \) by rearranging the identity: \[ \sin^2(\theta) = 1 - \cos^2(\theta) \] Thus, understanding and using the Pythagorean Identity is crucial in trigonometry exercises.

Second Quadrant

Angles in the coordinate plane are divided into four quadrants, with each quadrant having different sign rules for the trigonometric functions. Understanding which quadrant an angle lies in can help predict the signs of the sine, cosine, and tangent functions. In the second quadrant, angles range from \(90^{\circ}\) to \(180^{\circ}\). In this quadrant, the sign characteristics are as follows:

• \( \sin(\theta) > 0 \)
• \( \cos(\theta) < 0 \)
• \( \tan(\theta) < 0 \)

This is because the x-coordinate (cosine) is negative, while the y-coordinate (sine) is positive in this region. Knowing this information is very helpful in determining the behavior of trigonometric functions when solving problems.

Cosine Identity

The cosine of an angle \( \theta \), denoted as \( \cos(\theta) \), is an important trigonometric function representing the adjacent side over the hypotenuse in a right-angled triangle. When we work with negative values of cosine, like \( \cos(\theta) = -\frac{2}{7} \), it indicates that the angle is situated in a quadrant where cosine values are negative. From the students' perspective, it's crucial to remember that:

• The cosine value tells us about the horizontal stretch in terms of the angle.
• Negative cosine will place the angle in either the second or third quadrant.

When solving problems, the cosine identity (\( \cos^2(\theta) = 1 - \sin^2(\theta) \)) can be useful in finding the sine values by rearranging the Pythagorean Identity.

Sine Identity

The sine function, represented as \( \sin(\theta) \), correlates to the opposite side over the hypotenuse in a triangle. It plays a pivotal role in understanding angle measurements and solving trigonometric equations. A positive sine value, such as found in the second quadrant with \( \sin(\theta) > 0 \), indicates that we're dealing with a vertical stretch that is positive. To compute \( \sin(\theta) \) when \( \cos(\theta) \) is known, leverage the Pythagorean identity: \( \sin^2(\theta) = 1 - \cos^2(\theta) \). For instance, if \( \cos(\theta) = -\frac{2}{7} \), solving gives us \( \sin(\theta) = \frac{\sqrt{45}}{7} \). Remember: always pay attention to the sign depending on the quadrant. In such cases, trigonometric identities help balance the conceptual and computational aspects of problem-solving.