Problem 52 - Algebra and Trigonometry

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$ \tan \theta=-4, \quad \sin \theta>0 $$

Step-by-Step Solution

Answer

\(\sin \theta = \frac{4\sqrt{17}}{17}, \cos \theta = \frac{-\sqrt{17}}{17}\).

Step 1: Determine the Quadrant for \(\theta\)

Since \( \tan \theta = -4 \) and \( \sin \theta > 0 \), \( \theta \) must be in the quadrant where tangent is negative and sine is positive. This occurs in the second quadrant.

Step 2: Set Up a Right Triangle

In the second quadrant, the tangent of \( \theta \) is negative. We can use the identity \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). Let's set the opposite side to be 4 (since tangent is 4 in magnitude) and the adjacent side to be -1, because tangent is negative.

Step 3: Find the Hypotenuse Using Pythagorean Theorem

Use the Pythagorean theorem to find the hypotenuse: \[c^2 = 4^2 + (-1)^2 = 16 + 1 = 17 \]Therefore, \( c = \sqrt{17} \).

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

In a right triangle, \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{\sqrt{17}} \). Rationalizing the denominator gives: \[\sin \theta = \frac{4\sqrt{17}}{17} \]

Step 5: Calculate \( \cos \theta \)

In a right triangle, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-1}{\sqrt{17}} \). Rationalizing the denominator gives: \[\cos \theta = \frac{-\sqrt{17}}{17} \]

Step 6: Verification

Verify that the calculated values satisfy the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):\[\left(\frac{4\sqrt{17}}{17}\right)^2 + \left(\frac{-\sqrt{17}}{17}\right)^2 = \frac{16 \cdot 17}{289} + \frac{17}{289} = \frac{289}{289} = 1 \]

Key Concepts

Second Quadrant

The second quadrant of the Cartesian coordinate system is where angles range from 90° to 180°. Here, the sine function is positive, making it distinct from cosine and tangent, which are negative.

Understanding the second quadrant is crucial for trigonometry because it affects the sign of trigonometric functions. When diagnosing which quadrant an angle falls into, like in our exercise, start by noting any given signs of the trig functions. For instance:

• If \( \tan \theta = -4 \) and \( \sin \theta > 0 \), this tells us immediately that \( \theta \) is in the second quadrant because tangent is negative and sine is positive in this region.

The second quadrant is often encountered in exercises involving angles in standard position, helping students recognize patterns in trigonometric identities.

Right Triangle

A right triangle is fundamental in trigonometry, formed by one right angle (90°) and two acute angles (less than 90°). In any right triangle, the sides include the hypotenuse (the longest side across from the right angle) and the other two sides meet at the right angle.

In problems involving trigonometric functions, like the current exercise, right triangles help translate trigonometric ratios into side lengths. By establishing that \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = -4 \), we recognize it as a characteristic of a right triangle in the second quadrant and set the respective side lengths.

When constructing a right triangle from given trigonometric information:

• Remember: the opposite side is aligned with the angle of interest.
• The adjacent side lies next to the angle.
• The hypotenuse is always positive.

This structure facilitates transitioning between angular and side length contexts in geometric and trigonometric calculations.

Pythagorean Theorem

The Pythagorean theorem is a core principle in mathematics that relates the lengths of the sides in a right triangle. It states \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

In our exercise, after we identified the opposite and adjacent sides as 4 and -1 respectively, the hypotenuse was computed with this theorem. We executed: \(c^2 = 4^2 + (-1)^2 = 16 + 1 = 17\)giving us: \( c = \sqrt{17} \). This critical step of using the Pythagorean theorem transforms trigonometric expressions into tangible lengths used to find other trigonometric functions.

The Pythagorean theorem is not only useful for immediate calculation but also serves as a validation tool. We used it to ensure consistency in our trigonometric identities by rechecking with \( \sin^2 \theta + \cos^2 \theta = 1 \), confirming accuracy in results.