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Chapter 1: Problem 31

Determine whether each statement is true or false. $$ \cos 35^{\circ}<\cos 45^{\circ} $$

Short Answer

Expert verified
False - the statement is incorrect because \(\cos 35^{\circ}\) is greater than \(\cos 45^{\circ}\).

Step by step solution

01

Understanding Cosine Function Values

The cosine function in trigonometry is a decreasing function in the first quadrant, meaning as the angle increases from 0 to 90 degrees, the cosine value decreases. This means that for any two angles \(A\) and \(B\) such that \(A < B\), \(\cos A > \cos B\).
02

Compare the Angles

Identify the angles given in the problem: \(35^{\circ}\) and \(45^{\circ}\). Here, \(35^{\circ}\) is less than \(45^{\circ}\).
03

Analyze the Inequality Statement

The statement to evaluate is \(\cos 35^{\circ} < \cos 45^{\circ}\). Based on the property of cosine function in the first quadrant (explained in Step 1), since \(35^{\circ}\) is less than \(45^{\circ}\), \(\cos 35^{\circ}\) should actually be greater than \(\cos 45^{\circ}\).
04

Conclusion

Since the cosine of a smaller angle in the first quadrant is greater, the statement \(\cos 35^{\circ} < \cos 45^{\circ}\) is false.

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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, denoted as \(\cos \), deals with the relationship between the angles and sides of a right triangle. It is one of the six fundamental trigonometric functions and is essential in both geometry and calculus. This function specifically returns the ratio of the adjacent side over the hypotenuse. In terms of unit circle representation, cosine represents the x-coordinate of a point on the circle.
In the first quadrant, where the angle ranges from 0 to 90 degrees, the cosine function is a decreasing function:
  • As the angle increases, from 0 to 90 degrees, cosine values decrease.
  • At 0 degrees, \( \cos(0^\circ) = 1 \).
  • At 90 degrees, \( \cos(90^\circ) = 0 \).
Understanding the behavior of the cosine function across different quadrants is critical in solving inequalities and evaluating trigonometric expressions.
Angle Comparison
When comparing angles, especially within the context of trigonometric functions, the order or magnitude of the angles plays a crucial role in determining the function values. In our specific scenario, we are comparing 35 degrees and 45 degrees.
Since both angles are positioned in the first quadrant:
  • The function values, specifically cosine, will follow the property that smaller angles have larger cosine values.
  • This means for angles like 35 degrees and 45 degrees, we should note that "since 35 is less than 45," \( \cos(35^\circ) \) will be greater than \( \cos(45^\circ) \).
Remember that understanding angle comparison is not just about noticing which number is smaller or larger but also recognizing how this relates to trigonometric function behavior.
Trigonometry Quadrant Properties
To master trigonometric inequalities, it is vital to understand the properties of each quadrant in the coordinate plane. Trigonometric functions are defined as circular functions based on the unit circle, which is split into four quadrants.
Each quadrant has unique properties affecting the sine, cosine, and tangent functions:
  • In the first quadrant, all trigonometric values (sine, cosine, and tangent) are positive.
  • Cosine values decrease as the angle increases from 0 to 90 degrees.
Outside of the first quadrant, the signs of trigonometric functions change:
  • For example, in the second quadrant, cosine values are negative.
  • The trigonometry quadrant properties help in predicting the sign and behavior of trigonometric functions.
Knowing these properties allows students to solve inequalities efficiently by predicting function behavior and comparing function values for given angles.

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Most popular questions from this chapter

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \cos \theta \tan \theta=\sin \theta $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ (\cos \theta+1)(\cos \theta-1)=-\sin ^{2} \theta $$

Using your calculator and rounding your answers to the nearest hundredth, find the remaining trigonometric ratios of \(\theta\) based on the given information. $$ \sec \theta=-1.24 \text { and } \theta \in \mathrm{QII} $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ (\sin \theta-\cos \theta)^{2}-1=-2 \sin \theta \cos \theta $$

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ 1-\frac{\sin \theta}{\csc \theta}=\cos ^{2} \theta $$
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