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Magazine Chapter 20: Problem 21
Prove the given identities. $$\sin x \sec x=\tan x$$
Short Answer
Expert verified
The identity \( \sin x \sec x = \tan x \) is proven using trigonometric definitions and simplification.
Step by step solution
01
Recall Trigonometric Definitions
Before we start proving the identity, let's recall the definitions of the trigonometric functions involved:1. The sine function, \( \sin x \), is defined as the ratio of the opposite side to the hypotenuse in a right triangle.2. The secant function, \( \sec x \), is the reciprocal of the cosine function, or \( \sec x = \frac{1}{\cos x} \).3. The tangent function, \( \tan x \), is the ratio of the sine function to the cosine function, or \( \tan x = \frac{\sin x}{\cos x} \).
02
Express \(\sec x\) in Terms of Cosine
We know that \( \sec x = \frac{1}{\cos x} \). Therefore, we can rewrite the left side of the equation \( \sin x \sec x \) as \( \sin x \times \frac{1}{\cos x} \).
03
Simplify the Expression
Substitute the expression for \( \sec x \) into the equation:\[\sin x \times \frac{1}{\cos x} = \frac{\sin x}{\cos x}\]This is exactly the expression for \( \tan x \) because \( \tan x = \frac{\sin x}{\cos x} \).
04
Conclusion
Since both sides of the identity \( \sin x \sec x = \tan x \) simplify to the same expression \( \frac{\sin x}{\cos x} \), the identity is proven to be true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sine Function
The sine function is a cornerstone of trigonometry, primarily dealing with angles and sides of right triangles. It is denoted by \( \sin x \), where \( x \) represents the angle. In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
For example:
For example:
- If in a right triangle, the opposite side measures 3 units, and the hypotenuse measures 5 units, then \( \sin x = \frac{3}{5} = 0.6 \).
Secant Function
The secant function is less commonly used than sine or cosine but is still significant in trigonometry. It is denoted by \( \sec x \) and defined as the reciprocal of the cosine function, such that \( \sec x = \frac{1}{\cos x} \).
Its importance shines particularly in calculus and analytical geometry, where it helps solve more complex trigonometric equations.
Understanding \( \sec x \) includes recognizing:
Its importance shines particularly in calculus and analytical geometry, where it helps solve more complex trigonometric equations.
Understanding \( \sec x \) includes recognizing:
- The secant function approaches infinity as the cosine of an angle approaches zero.
- Secant values are often greater than or equal to 1, as long as the cosine is between -1 and 1.
Tangent Function
The tangent function acts as a bridge between sine and cosine, defining a relationship between these two basic trigonometric functions. The tangent of an angle \( x \) is represented as \( \tan x \), calculated by the formula \( \tan x = \frac{\sin x}{\cos x} \).
This function is particularly useful in surveying, navigation, and in the study of periodic functions.
Important features of the tangent function include:
This function is particularly useful in surveying, navigation, and in the study of periodic functions.
Important features of the tangent function include:
- Its periodic nature, repeating every \( \pi \) radians.
- It becomes undefined whenever \( \cos x = 0 \), which corresponds to angles where \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
Trigonometric Definitions
Trigonometric definitions lay the foundation for understanding and applying various trigonometric identities. They express relationships between the angles and sides of triangles.
Key definitions include:
Key definitions include:
- **Sine**: \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \)
- **Cosine**: \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \)
- **Tangent**: \( \tan x = \frac{\sin x}{\cos x} \)
- **Secant**: \( \sec x = \frac{1}{\cos x} \)
