Problem 13 - Algebra and Trigonometry

Question

Use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\sin \theta=-\frac{9}{41}, \theta\) lies in quadrant III.

Step-by-Step Solution

Answer

Finally, we have the values as \(\sin 2 \theta = \frac{720}{1681}\), \(\cos 2 \theta = -\frac{639}{1681}\), and \(\tan 2 \theta = -\frac{720}{639}\)

Step 1: Calculating \(\cos \theta\)

In the third quadrant, sine is negative and cosine is also negative. By using Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), solve for \(\cos \theta\):\[-(\sqrt{1-\left(-\frac{9}{41}\right)^2}) = -\frac{40}{41}\]

Step 2: Calculate \(\sin 2 \theta\)

Now, use the double angle formula for sine: \[\sin 2 \theta = 2 \sin \theta \cos \theta.\] Substitute given \(\sin \theta = -\frac{9}{41}\) and calculated \(\cos \theta = -\frac{40}{41}\), \[\sin 2 \theta = 2 \left(-\frac{9}{41}\right) \left(-\frac{40}{41}\right) = \frac{720}{1681}\]

Step 3: Calculate \(\cos 2 \theta\)

Use the double angle formula for cosine: \[\cos 2 \theta = \cos^2 \theta - \sin^2 \theta.\] Substitute given \(\sin \theta = -\frac{9}{41}\) and calculated \(\cos \theta = -\frac{40}{41}\), \[\cos 2 \theta = \left(-\frac{40}{41}\right)^2 - \left(-\frac{9}{41}\right)^2 = -\frac{639}{1681}\]

Step 4: Calculate \(\tan 2 \theta\)

Obtain \(\tan 2 \theta\) by dividing \(\sin 2 \theta\) by \(\cos 2 \theta\). Substitute \(\sin 2 \theta = \frac{720}{1681}\) and \(\cos 2 \theta = -\frac{639}{1681}\),\[\tan 2 \theta = \frac{\sin 2 \theta}{\cos 2 \theta} = -\frac{720}{639}\]

Key Concepts

Understanding Double Angle Formulas

Double angle formulas are essential in trigonometry for simplifying expressions and solving equations involving multiple angles. These formulas allow us to express trigonometric functions of twice an angle in terms of trigonometric functions of the original angle. For the sine and cosine functions, the formulas are:

• \( \sin 2 \theta = 2 \sin \theta \cos \theta \)
• \( \cos 2 \theta = \cos^2 \theta - \sin^2 \theta \)

The double angle formula for sine shows a direct relation between \( \sin 2 \theta \) and the product of \( \sin \theta \) and \( \cos \theta \). On the other hand, for cosine, it is expressed as the difference of squares of the cosine and sine of \( \theta \). Understanding these formulas helps you to transform and simplify expressions, especially in integration and solving trigonometric equations.
These formulas are derived from basic trigonometric identities, and they provide a powerful tool to simplify the calculations as shown in the step-by-step solution of this problem.

Trigonometry in Quadrants

Understanding how trigonometric functions behave in different quadrants is crucial for solving problems involving angles. The four quadrants in a coordinate plane define the sign of each trigonometric function. Each quadrant affects the value of sine, cosine, and tangent differently:

• In Quadrant I, all functions are positive.
• In Quadrant II, sine is positive, but cosine and tangent are negative.
• In Quadrant III, sine and cosine are negative, but tangent is positive.
• In Quadrant IV, cosine is positive, while sine and tangent are negative.

This knowledge helps to determine the correct signs when calculating trigonometric values. Since the angle \( \theta \) in our problem is in Quadrant III, \( \sin \theta \) and \( \cos \theta \) both take negative values, directly affecting the outcome of trigonometric calculations using double angle formulas.

The Pythagorean Identity

One of the fundamental identities in trigonometry is the Pythagorean identity, which is derived from the Pythagorean theorem. It states:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is essential when you need to find either the sine or cosine of an angle, given the other, especially if one is negative, as in our problem.
The Pythagorean identity can be manipulated to isolate either sine or cosine, allowing you to solve for unknown values. In this exercise, we used it to find \( \cos \theta \) from a known \( \sin \theta \), by rearranging it to:\[ \cos \theta = \pm\sqrt{1 - \sin^2 \theta} \]Given that \( \theta \) is in Quadrant III, where cosine is negative, this information was critical in selecting the correct sign for \( \cos \theta \), ensuring the calculated values align with the quadrant’s rules.

Angle Calculations

Calculating angles in trigonometry typically involves using identities and understanding quadrant-specific behavior. Besides finding the double angles as discussed, you may encounter tasks requiring you to manipulate inverse trigonometric functions or determine angles that fit specific criteria.
Angle calculations often involve determining reference angles or complementary angles to simplify problems. For instance, an angle in the third quadrant can be related back to its reference angle in the first quadrant:

• Reference angle is the smallest angle formed with the x-axis.
• It helps in calculating accurate trigonometric values while respecting the sign determined by the quadrant.

These insights are commonly paired with formulas like the double angle or Pythagorean identities, allowing you to solve complex problems systematically and confidently, fully embracing how trigonometric rules shape angle calculations.