Problem 103
Question
Let \(s\) be a real number corresponding to the point ( \(a, b\) ) on the unit circle. Use this information to determine an expression representing the sine and cosine of each real number. $$-s$$
Step-by-Step Solution
Verified Answer
The sine and cosine of \\( -s \\) are \\( -b \\) and \\( a \\) respectively.
1Step 1: Understand the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin \(0, 0\) in the coordinate plane. Points on the unit circle have coordinates that can be expressed as \( (\cos(\theta), \sin(\theta)) \.\) In this problem, \( (a, b) \) represent \( (\cos(s), \sin(s)) \.\) Therefore, \( a = \cos(s) \) and \( b = \sin(s) \.\)
2Step 2: Apply Even-Odd Identities in Trigonometry
Recall that cosine is an even function and sine is an odd function. This means that \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \.\) We can use these identities to find \( \cos(-s) \) and \( \sin(-s) \.\)
3Step 3: Compute Cosine and Sine for -s
Using the even-odd identities, we find that \( \cos(-s) = \cos(s) = a \) and \( \sin(-s) = -\sin(s) = -b \.\) This means the point \( (-s) \) on the unit circle corresponds to \( (a, -b) \.\)
Key Concepts
TrigonometryEven-Odd IdentitiesSine and Cosine Functions
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. A key concept in trigonometry is its relationship with the unit circle. The unit circle makes it easier to understand how the sine, cosine, and other trigonometric functions behave for different angles (or values of \(s\)).
In a coordinate system, the unit circle is centered at the origin \((0, 0)\) and has a radius of 1. Any point \((a, b)\) on this circle represents the coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
Understanding the unit circle is crucial in trigonometry because it helps explain trigonometric functions:
In a coordinate system, the unit circle is centered at the origin \((0, 0)\) and has a radius of 1. Any point \((a, b)\) on this circle represents the coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
Understanding the unit circle is crucial in trigonometry because it helps explain trigonometric functions:
- Sine (\(\sin\)) measures the vertical distance from the x-axis to a point on the unit circle.
- Cosine (\(\cos\)) measures the horizontal distance from the y-axis to a point on the unit circle.
Even-Odd Identities
In trigonometry, some functions are classified as even or odd. Recognizing these patterns is essential for simplifying expressions and solving problems.
**Even Functions**
Even functions have the property that \(\cos(-\theta) = \cos(\theta)\). Geometrically, this means that flipping a point along the y-axis doesn't change the x-coordinate's behavior. Hence, cosine is an even function. It's symmetric about the y-axis.
**Odd Functions**
Odd functions, such as sine, have the property that \(\sin(-\theta) = -\sin(\theta)\). For these functions, flipping over the y-axis means the y-coordinate becomes its own opposite. Thus, sine is an odd function, symmetric about the origin.
This knowledge simplifies calculations and allows us to quickly determine how the functions behave under transformations, like flipping or rotating angles.
**Even Functions**
Even functions have the property that \(\cos(-\theta) = \cos(\theta)\). Geometrically, this means that flipping a point along the y-axis doesn't change the x-coordinate's behavior. Hence, cosine is an even function. It's symmetric about the y-axis.
**Odd Functions**
Odd functions, such as sine, have the property that \(\sin(-\theta) = -\sin(\theta)\). For these functions, flipping over the y-axis means the y-coordinate becomes its own opposite. Thus, sine is an odd function, symmetric about the origin.
This knowledge simplifies calculations and allows us to quickly determine how the functions behave under transformations, like flipping or rotating angles.
Sine and Cosine Functions
The sine and cosine functions are fundamental to trigonometry and describe circular motion.
**Sine Function**
The sine of an angle \(\theta\), written \(\sin(\theta)\), is the y-coordinate of a point on the unit circle. As the angle increases, the sine values change smoothly, reaching a maximum of 1, minimum of -1, and crossing zero at multiple points.
**Cosine Function**
The cosine of an angle \(\theta\), written \(\cos(\theta)\), is similarly the x-coordinate of a point on the unit circle. It moves between values of 1 and -1 and shares a similar periodic behavior to sine but shifted by 90 degrees. This shift means that when cosine is at a maximum, sine is at zero, and vice versa.
**Sine Function**
The sine of an angle \(\theta\), written \(\sin(\theta)\), is the y-coordinate of a point on the unit circle. As the angle increases, the sine values change smoothly, reaching a maximum of 1, minimum of -1, and crossing zero at multiple points.
**Cosine Function**
The cosine of an angle \(\theta\), written \(\cos(\theta)\), is similarly the x-coordinate of a point on the unit circle. It moves between values of 1 and -1 and shares a similar periodic behavior to sine but shifted by 90 degrees. This shift means that when cosine is at a maximum, sine is at zero, and vice versa.
- Both functions are periodic, repeating every \(2\pi\) radians.
- Their understanding is critical for problems involving harmonic motion, waves, and various engineering tasks.
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