Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of \(\sin 2 u, \cos 2 u\) and tan \(2 u\) using the double- angle formulas. $$\sin u=-\frac{3}{5}, \quad 3 \pi / 2

Step-by-Step Solution

Answer

The exact values of \(\sin 2u\), \(\cos 2u\), and \(\tan 2u\) using the given double angle formulas with \(\sin u = -3/5\) in the range \(3 \pi / 2

Step 1: Determine the value of \(\cos u\)

We know that \(\sin u = -3/5\) lies in the fourth quadrant. The formula \(\cos^2u = 1 - \sin^2u\), derived from the Pythagorean identity, is used to compute the value of \(\cos u\). Since this is the fourth quadrant, \(\cos u\) is positive, so \(\cos u = \sqrt{1 - (-3/5)^2} = 4/5\).

Step 2: Apply the double-angle formula for \(\sin 2u\)

Substitute the values of \(\sin u\) and \(\cos u\) into the double-angle formula \(\sin 2u = 2sin(u)cos(u)\). Therefore, \(\sin 2u = 2*(-3/5)*(4/5) = -24/25\).

Step 3: Apply the double-angle formula for \(\cos 2u\)

Use the double angle identity for \(\cos 2u = cos^2(u) - sin^2(u)\) and insert the values of \(\sin u\) and \(\cos u\) to get \(\cos 2u = (4/5)^2 - (-3/5)^2 = 7/25\).

Step 4: Find the value for \(\tan 2u\)

\(\tan 2u\) is the ratio of \(\sin 2u\) to \(\cos 2u\), i.e., \(\tan 2u = sin(2u) / cos(2u) = -24/25 / 7/25 = -24/7\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle, the sum of the squares of sine and cosine equals one: \[\sin^2 u + \cos^2 u = 1\]This identity helps us connect the values of sine and cosine. If one is known, the other can be found. In our exercise, we used it to find \(\cos u\) when \(\sin u\) was known. To do this:

• Calculate \(\sin^2 u\), which is \((-3/5)^2 = 9/25\).
• Subtract \(9/25\) from 1 to find \(\cos^2 u = 16/25\).
• Take the square root to get \(\cos u = 4/5\).\(\cos u\) is positive in the fourth quadrant.

This identity is invaluable for solving many trigonometric problems.

Trigonometry

Trigonometry is the study of angles and their relationships. It uses functions like sine, cosine, and tangent to describe these relationships. Each trigonometric function helps measure different ratios of sides in a right triangle. For example:

• \(\sin u\) is the ratio of the opposite side to the hypotenuse.
• \(\cos u\) is the ratio of the adjacent side to the hypotenuse.
• \(\tan u\) is the ratio of the opposite side to the adjacent side.

In this exercise, we worked in the context of standard trigonometric circle coordinates, using the values of \(\sin u\) and \(\cos u\) to find double-angle results. Understanding these basics is crucial to tackling more complex problems.

Sin 2u

The double-angle formula for sine, \(\sin 2u = 2 \sin u \cos u\), allows us to calculate the sine of double angles using known values of single angles. In the exercise:

• We have \(\sin u = -3/5\) and \(\cos u = 4/5\).
• Applying the formula, \(\sin 2u = 2 (-3/5) (4/5) = -24/25\).

This method efficiently gives the sine of a double angle, which could otherwise be complex to determine directly. It showcases the versatility and usefulness of trigonometric formulas.

Cos 2u

The double-angle formula for cosine provides multiple ways to calculate double angles. One commonly used is \(\cos 2u = \cos^2 u - \sin^2 u\). Here's how we calculated it in the exercise:

• Using \(\cos u = 4/5\) and \(\sin u = -3/5\).
• Substitute into the formula: \(\cos 2u = (4/5)^2 - (-3/5)^2\).
• Simplify: \(\cos 2u = 16/25 - 9/25 = 7/25\).

This result highlights how double-angle formulas streamline finding double-angle trigonometric values, saving time and effort.

Tan 2u

The double-angle formula for tangent uses both \(\sin 2u\) and \(\cos 2u\) values. It's given by the ratio \(\tan 2u = \frac{\sin 2u}{\cos 2u}\).To solve this in our exercise:

• We have \(\sin 2u = -24/25\) and \(\cos 2u = 7/25\).
• Calculate \(\tan 2u = -24/25 \div 7/25\).
• Divide: \(\tan 2u = -24/7\).

This formula helps in quickly finding tangent values for double angles by using previously calculated sine and cosine values. It’s a powerful tool in trigonometry.

Previous

Problem 20

Next

Problem 21

Other exercises in this chapter

Problem 20

Verify the identity. $$\tan ^{2} y\left(\csc ^{2} y-1\right)=1$$

View solution

Problem 20

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x=\frac{\sqrt{3}}{2}$$

View solution

Problem 21

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

View solution

Problem 21

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}

View solution

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that for any angle, the sum of the squares of sine and cosine equals one: \[\sin^2 u + \cos^2 u = 1\]This identity helps us connect the values of sine and cosine. If one is known, the other can be found. In our exercise, we used it to find \(\cos u\) when \(\sin u\) was known. To do this:

• Calculate \(\sin^2 u\), which is \((-3/5)^2 = 9/25\).
• Subtract \(9/25\) from 1 to find \(\cos^2 u = 16/25\).
• Take the square root to get \(\cos u = 4/5\).\(\cos u\) is positive in the fourth quadrant.

This identity is invaluable for solving many trigonometric problems.

Trigonometry

Trigonometry is the study of angles and their relationships. It uses functions like sine, cosine, and tangent to describe these relationships. Each trigonometric function helps measure different ratios of sides in a right triangle. For example:

• \(\sin u\) is the ratio of the opposite side to the hypotenuse.
• \(\cos u\) is the ratio of the adjacent side to the hypotenuse.
• \(\tan u\) is the ratio of the opposite side to the adjacent side.

In this exercise, we worked in the context of standard trigonometric circle coordinates, using the values of \(\sin u\) and \(\cos u\) to find double-angle results. Understanding these basics is crucial to tackling more complex problems.

Sin 2u

The double-angle formula for sine, \(\sin 2u = 2 \sin u \cos u\), allows us to calculate the sine of double angles using known values of single angles. In the exercise:

• We have \(\sin u = -3/5\) and \(\cos u = 4/5\).
• Applying the formula, \(\sin 2u = 2 (-3/5) (4/5) = -24/25\).

This method efficiently gives the sine of a double angle, which could otherwise be complex to determine directly. It showcases the versatility and usefulness of trigonometric formulas.

Cos 2u

The double-angle formula for cosine provides multiple ways to calculate double angles. One commonly used is \(\cos 2u = \cos^2 u - \sin^2 u\). Here's how we calculated it in the exercise:

• Using \(\cos u = 4/5\) and \(\sin u = -3/5\).
• Substitute into the formula: \(\cos 2u = (4/5)^2 - (-3/5)^2\).
• Simplify: \(\cos 2u = 16/25 - 9/25 = 7/25\).

This result highlights how double-angle formulas streamline finding double-angle trigonometric values, saving time and effort.

Tan 2u

The double-angle formula for tangent uses both \(\sin 2u\) and \(\cos 2u\) values. It's given by the ratio \(\tan 2u = \frac{\sin 2u}{\cos 2u}\).To solve this in our exercise:

• We have \(\sin 2u = -24/25\) and \(\cos 2u = 7/25\).
• Calculate \(\tan 2u = -24/25 \div 7/25\).
• Divide: \(\tan 2u = -24/7\).

This formula helps in quickly finding tangent values for double angles by using previously calculated sine and cosine values. It’s a powerful tool in trigonometry.