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Chapter 6: Problem 46

Simplify each expression. \(\cos (y)+\cos (-y)\)

Short Answer

Expert verified
2 \cos(y)

Step by step solution

01

- Understand cosine's properties

Recall that the cosine function is an even function. This means that \(\cos(-y) = \cos(y)\).
02

- Substitute using cosine's property

Substitute \(\cos(-y)\) with \(\cos(y)\) in the original expression. Thus, the expression \(\cos (y)+\cos (-y)\) becomes \(\cos (y)+\cos (y)\).
03

- Combine like terms

Combine the terms \(\cos (y)+\cos (y)\) to get \(\2 \cos(y)\).

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Key Concepts

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

Cosine Function
The cosine function is one of the fundamental trigonometric functions. It's often abbreviated as 'cos'. The function takes an angle and returns a value between -1 and 1. The cosine function is particularly useful in various fields such as physics, engineering, and computer graphics. In the unit circle, it represents the x-coordinate of a point on the circle.

Cosine has a periodicity of 360 degrees or 2蟺 radians, meaning \(\cos (\theta + 2\pi) = \cos (\theta)\). This periodic property makes it predictable and consistent. The cosine of special angles like 0掳, 90掳, 180掳, and 270掳 are typically used in trigonometry and calculus problems.
Even Functions
Even functions are symmetric with respect to the y-axis. Mathematically, a function f(x) is even if and only if \({f(-x) = f(x)}\). For the cosine function, \(\cos(-y) = \cos(y)\).

Because cosine is an even function, we can simplify expressions involving \(\cos(-y)\) by simply substituting \(\cos(y)\). This property can save a lot of time during problem-solving and helps in simplifying trigonometric identities.
Simplifying Expressions
Simplifying expressions is the process of making them easier to understand or solve. In trigonometry, this often involves using properties of trigonometric functions, like their even or odd nature, or periodicity.

For example, in the expression \(\cos (y)+e \cos (-y)\), knowing that cosine is an even function allows us to write it as \(\cos (y)\) + \(\cos(y)\). Simplified expressions make it easier to analyze and solve problems, reducing the complexity of equations.
Combining Like Terms
Combining like terms involves adding or subtracting terms containing the same variable and exponent. This is a fundamental step in simplifying algebraic and trigonometric expressions.

For \(\cos (y) + \cos (y)\), we're effectively adding the same term together. So, combining like terms, we get \({2 \cos(y)}\).

Combining like terms helps in streamlining equations, making them easier to work with and solve. This is particularly crucial both in algebra and trigonometry for solving equations efficiently.

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Most popular questions from this chapter

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$\cot ^{2} \theta-4 \cot \theta+2=0$$

Find all values of \(\alpha\) in degrees that satisfy each equation. Round approximate answers to the nearest tenth of a degree. $$\sin 3 \alpha=0.34$$

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$\frac{\tan 3 \theta-\tan \theta}{1+\tan 3 \theta \tan \theta}=\sqrt{3}$$

Solve each problem. Find the exact value of \(\sin (2 \alpha)\) given that \(\tan (\alpha)=-8 / 15\) and \(\alpha\) is in quadrant IV.

One way to solve an equation with a graphing calculator is to rewrite the equation with 0 on the right-hand side, then graph the function that is on the left-hand side. The x-coordinate of each \(x\) -intercept of the graph is a solution to the original equation. For each equation, find all real solutions (to the nearest tenth) in the interval \([0,2 \pi).\) $$x^{2}=\sin x$$
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