Problem 4 - Algebra and Trigonometry

Question

\(3-10\) Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$ \tan x=-\frac{4}{3}, \quad x \text { in Quadrant II } $$

Step-by-Step Solution

Answer

\(\sin 2x = \frac{24}{25}, \cos 2x = -\frac{7}{25}, \tan 2x = -\frac{24}{7}\).

Step 1: Identify Reference Triangle

Given that \(\tan x = -\frac{4}{3}\) and \(x\) in Quadrant II, draw a right triangle for \(x\) with opposite side \(-4\) and adjacent side \(3\). The hypotenuse can be found using the Pythagorean theorem: \(c = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = 5\).

Step 2: Determine Sine and Cosine

From the triangle, we have \(\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-4}{5}\) and \(\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}\). However, in Quadrant II, \(\cos x\) should be negative, so \(\cos x = -\frac{3}{5}\).

Step 3: Apply Double Angle Formulas for Sine and Cosine

Use the formulas \(\sin 2x = 2 \sin x \cos x\) and \(\cos 2x = \cos^2 x - \sin^2 x\). Substitute \(\sin x = -\frac{4}{5}\) and \(\cos x = -\frac{3}{5}\) to get:\[\sin 2x = 2 \left(\frac{-4}{5}\right) \left(\frac{-3}{5}\right) = \frac{24}{25}\]\[\cos 2x = \left(\frac{-3}{5}\right)^2 - \left(\frac{-4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25}\]

Step 4: Calculate the Tangent of Double Angle

Use the formula \(\tan 2x = \frac{\sin 2x}{\cos 2x}\). Substitute the values obtained:\[\tan 2x = \frac{\frac{24}{25}}{-\frac{7}{25}} = \frac{24}{25} \times \frac{-25}{7} = -\frac{24}{7}\]

Step 5: Conclusion and Summary

The values of the trigonometric functions for the double angle are determined. We found \(\sin 2x = \frac{24}{25}\), \(\cos 2x = -\frac{7}{25}\), and \(\tan 2x = -\frac{24}{7}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides.
They are fundamental in trigonometry and are particularly useful when dealing with right-angled triangles. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).

• The sine of an angle, \(\sin x\), is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle, \(\cos x\), is the ratio of the length of the adjacent side to the hypotenuse.
• The tangent of an angle, \(\tan x\), is the ratio of the length of the opposite side to the adjacent side.

Double angle identities are useful for finding the sine, cosine, and tangent of twice an angle. These formulas include:

• \(\sin 2x = 2\sin x\cos x\)
• \(\cos 2x = \cos^2 x - \sin^2 x\)
• \(\tan 2x = \frac{2\tan x}{1-\tan^2 x}\)

Knowing these identities allows you to explore how the angle's properties change when doubled.

Reference Triangle

A reference triangle is a right triangle drawn to help determine the trigonometric values of an angle.
It's especially useful in determining the signs and values of trigonometric ratios in specific quadrants. In this problem, we made use of a reference triangle to find the hypotenuse when the opposite and adjacent sides are known.
Here’s how you go about it:

• When given \(\tan x = -\frac{4}{3}\), it means the opposite side is \(-4\) and the adjacent side is \(3\).
• Since the angle is in the second quadrant, both the opposite side can be negative and hypotenuse will be positive.
• The hypotenuse is found using the Pythagorean theorem: \[c = \sqrt{(-4)^2 + 3^2} = 5\].

Reference triangles simplify the process of understanding and applying trigonometric identities by anchoring the angles and their ratios to tangible values.

Quadrants

In trigonometry, the coordinate plane is divided into four sections called quadrants.
Each quadrant corresponds to a range of angle measurements:

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

For the given problem, knowing that \(x\) is in Quadrant II helps us determine that \(\sin x\) must be positive and \(\cos x\) negative.
This knowledge is crucial as it guides the calculation of double angles and the respective trigonometric functions.