Problem 88 Show that $$ \cos (\pi-\thet... [FREE SOLUTION]

Chapter 4: Problem 88

Show that $$ \cos (\pi-\theta)=-\cos \theta $$ for every angle $\theta$.

Short Answer

Expert verified

Using the cosine difference formula, $\cos(\pi - \theta) = \cos \pi \cos \theta + \sin \pi \sin \theta$. Since $\cos \pi = -1$ and $\sin \pi = 0$, we get $\cos(\pi - \theta) = (-1) \cos \theta + (0) \sin \theta = -\cos \theta$.

Step by step solution

Recall the cosine difference formula

The cosine difference formula is: \[ \cos (A - B) = \cos A \cos B + \sin A \sin B \]. In this exercise, we have $A = \pi$ and $B = \theta $.

Use the cosine difference formula with our angles

From Step 1, substitute $A = \pi$ and $B = \theta $ into the formula: \[ \cos (\pi - \theta) = \cos \pi \cos \theta + \sin \pi \sin \theta \]

Calculate the sine and cosine of 蟺

Recall that $\cos \pi = -1$ and $\sin \pi = 0$. Substitute these values into the formula from Step 2: \[ \cos (\pi - \theta) = (-1) \cos \theta + (0) \sin \theta \]

Simplify and finish

Since the $\sin \theta$ term is multiplied by zero, it will not contribute to the result. Therefore, we are left with: \[ \cos (\pi - \theta) = -\cos \theta \] As we aimed to show, the cosine of the difference between 蟺 and 胃 equals the negative of the cosine of 胃.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Difference Formula

The cosine difference formula is a key identity in trigonometry that simplifies the calculation of the cosine of the difference between two angles. This formula is written as: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] Here, $A$ and $B$ are the angles for which you want to determine the cosine of their difference. By breaking down the expression into cosine and sine components, this formula allows us to handle complex angle computations more easily.

**Cos**: Represents the cosine function, describing the relation of an angle to a right triangle's side lengths.
**Sin**: Short for sine, another function crucial for trigonometric calculations.

Using this identity becomes particularly useful when verifying other trigonometric equations or simplifying expressions, as it enables manipulation of the components to reach the desired form.

Angle Subtraction

Angle subtraction deals with finding the trigonometric functions of the difference between two angles, such as $ \pi - \theta $ in our example. This concept is crucial for many physics and engineering problems, where angle differences arise naturally. Here's how it works:

When you subtract an angle $ \theta $ from $ \pi $, you essentially reflect the angle across the x-axis on the unit circle.
This reflection is why $ \cos(\pi - \theta) = -\cos \theta $; the x-component, which cosine represents, is inverted.

Breaking down angles into their components through subtraction helps in various applications, such as determining direction, or when working in non-standard orientation systems.

Cosine Function

The cosine function, denoted as $\cos$, is fundamental in trigonometry. It relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. But it goes beyond triangles, becoming essential in wave analysis, circular motion, and oscillations.

**Definition**: $ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} $
**Properties**: It's periodic with a period of $2\pi$ and symmetric around the y-axis.
**Values**: For special angles $0, \frac{\pi}{2}, \pi,$ etc., these values are well-known and useful for proofs and problem-solving.

Understanding the cosine function's behavior is crucial for identifying patterns in problems involving periodic or cyclic phenomena.

Trigonometry Proofs

Trigonometry proofs require a deep understanding of identities and a creative approach to algebraic manipulation. Proving statements like $ \cos(\pi - \theta) = -\cos \theta $ involves recognizing known identities and applying them effectively. Steps include:

**Starting with Known Identities**: Begin with fundamental equations like the cosine difference formula.
**Substituting Values**: Replace variables with specific angles or known trigonometric values.
**Simplifying Expressions**: Use algebra to simplify and rearrange expressions to achieve the goal of the proof.

Mastering trigonometry proofs not only enhances problem-solving skills but also provides insights into the interconnectedness of mathematical concepts. Engaging with proofs sharpens logical thinking and strengthens the foundation for more advanced studies.

91影视

Show that $$ \cos (\pi-\theta)=-\cos \theta $$ for every angle \(\theta\).

Short Answer

Step by step solution

Recall the cosine difference formula

Use the cosine difference formula with our angles

Calculate the sine and cosine of 蟺

Simplify and finish

Key Concepts

Cosine Difference Formula

Angle Subtraction

Cosine Function

Trigonometry Proofs

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