Problem 102 - Precalculus Essentials

Question

In Exercises \(102-105,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. The double-angle identities are derived from the sum identities by adding an angle to itself.

Step-by-Step Solution

Answer

The statement that the double-angle identities are derived from the sum identities by adding an angle to itself makes partial sense: it is correct for the sine function, but not wholly correct for the cosine function

Step 1: State the Double-Angle and Sum Identities

The double-angle identities are: \1. \(\sin(2\Theta)=2\sin(\Theta)\cos(\Theta)\) \2. \(\cos(2\Theta)=\cos^2(\Theta)-\sin^2(\Theta)\) \And, the sum identities are: \1. \(\sin(\Theta+\phi) = \sin(\Theta)\cos(\phi) + \cos(\Theta)\sin(\phi)\) \2. \(\cos(\Theta+\phi) = \cos(\Theta)\cos(\phi) - \sin(\Theta)\sin(\phi)\)

Step 2: Express Double-Angle Identities in terms of Sum Identities

For \(\sin(2\Theta)\), if you substitute \(\Theta+\Theta\) into the sum identity for sine, you get: \(\sin(\Theta+\Theta) = \sin(\Theta) cos(\Theta) + cos(\Theta) sin(\Theta) = 2 \sin(\Theta) cos(\Theta)\), which is the double-angle identity for sine. Although, for \(\cos(2\Theta)\), substituting \(\Theta+\Theta\) into the sum identity for cosine doesn't give us the exact double-angle identity for cosine.

Step 3: Analyze the given statement

The statement mentioned asserts that the double-angle identities are derived from sum identities by adding an angle to itself. As seen in Step 2, this is true for the sine function, but not exactly the case for cosine. Therefore, the given statement partially makes sense.

Key Concepts

sum identities

Sum identities play a crucial role in trigonometry by allowing us to calculate the sine and cosine of the sum of two angles. This is essential for solving various types of problems, especially when dealing with combined angles.
Sum identities are given as follows:

• For sine: \( \sin(\Theta+\phi) = \sin(\Theta)\cos(\phi) + \cos(\Theta)\sin(\phi) \)
• For cosine: \( \cos(\Theta+\phi) = \cos(\Theta)\cos(\phi) - \sin(\Theta)\sin(\phi) \)

These identities allow us to break down complex trigonometric expressions into simpler, more manageable parts. By understanding and applying these identities, you can solve angles that would otherwise require more cumbersome calculations.
Moreover, they form the basis for deriving other important trigonometric identities, such as the double-angle identities.

trigonometric identities

Trigonometric identities form the backbone of many mathematical computations involving angles and triangles. They are equations that hold true for all values of the variables involved.
These identities include fundamental relations involving sine, cosine, and tangent, among others. They help simplify expressions and solve complex trigonometric equations, and are crucial in calculus, physics, and engineering.
Some key benefits of understanding trigonometric identities include:

• Solving complex equations quickly.
• Facilitating the integration and differentiation of trigonometric functions in calculus.
• Enabling the derivation of further useful identities like the power-reduction identities and half-angle identities.

Familiarity with these identities not only enhances computational efficiency but also deepens understanding of geometric and periodic phenomena.

sine and cosine

Sine and cosine are foundational trigonometric functions that describe a relationship between the angles and sides of a right triangle. In the unit circle perspective, these functions effectively map an angle to coordinates on the circle.
The sine function is defined as the y-coordinate, while the cosine function is the x-coordinate of a point on the unit circle that corresponds to a given angle.
Here are a few critical points about sine and cosine:

• The maximum and minimum values of sine and cosine are 1 and -1, respectively.
• Both are periodic functions with a period of \(2\pi\).
• They have key roles in defining other trigonometric identities and concepts, such as angle addition and double-angle identities.

Understanding sine and cosine is vital for mastering trigonometry, as they are used to describe waves and oscillations, model real-world phenomena, and are integral in advanced mathematics topics.

angle addition

The angle addition formulas allow us to express the trigonometric functions of the sum of two angles in terms of the functions of the individual angles. These are critical in various fields that apply trigonometry extensively.
For any two angles \(\Theta\) and \(\phi\), the formulas are:

• \(\sin(\Theta+\phi) = \sin(\Theta)\cos(\phi) + \cos(\Theta)\sin(\phi)\)
• \(\cos(\Theta+\phi) = \cos(\Theta)\cos(\phi) - \sin(\Theta)\sin(\phi)\)

These formulas are instrumental in many mathematical procedures, including proving the double-angle identities where an angle is added to itself.
By increasing comprehension of angle addition, one gains the ability to manipulate and solve more complex trigonometric problems, enhancing understanding in both theoretical and applied mathematics contexts.