1.2 Review of Trigonometry
Trigonometry plays a fundamental role in calculus, particularly when dealing with periodic functions, derivatives, and integrals involving angles. In this section, we will review the key concepts of trigonometry, including trigonometric functions, identities, and their applications.
1.2.1 Angles and Their Measurement
An angle is a measure of rotation between two rays. Angles can be measured in degrees or radians.
- Degrees: A full rotation is \(360^\circ\).
- Radians: A full rotation is \(2\pi) radians. Thus, (360^\circ = 2\pi\) radians, and \(180^\circ = \pi\) radians.
To convert between degrees and radians:
\[
\text{Radians} = \frac{\pi}{180^\circ} \times \text{Degrees}, \quad \text{Degrees} = \frac{180^\circ}{\pi} \times \text{Radians}
\]
1.2.2 The Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The angle \( \theta \), measured in radians, corresponds to a point on the circle at \( (x, y) \), where:
\[
x = \cos(\theta), \quad y = \sin(\theta)
\]
Thus, any point on the unit circle can be described as \( (\cos(\theta), \sin(\theta)) \).
Some important angle values (in radians) and their corresponding coordinates are:
- \( \theta = 0 \): \( (\cos(0), \sin(0)) = (1, 0) \)
- \( \theta = \frac{\pi}{2} \): \( (\cos\left(\frac{\pi}{2}\right), \sin\left(\frac{\pi}{2}\right)) = (0, 1) \)
- \( \theta = \pi \): \( (\cos(\pi), \sin(\pi)) = (-1, 0) \)
- \( \theta = \frac{3\pi}{2} \): \( (\cos\left(\frac{3\pi}{2}\right), \sin\left(\frac{3\pi}{2}\right)) = (0, -1) \)
1.2.3 Trigonometric Functions
The six basic trigonometric functions are based on the ratios of the sides of a right triangle or the coordinates on the unit circle.
For a right triangle with angle \( \theta \), adjacent side \( a \), opposite side \( b \), and hypotenuse \( c \), the trigonometric functions are defined as:
- Sine:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{b}{c}
\] - Cosine:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{c}
\] - Tangent:
\[
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{b}{a}
\] - Cosecant:
\[
\csc(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{c}{b}
\] - Secant:
\[
\sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{c}{a}
\] - Cotangent:
\[
\cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{a}{b}
\]
1.2.4 Fundamental Trigonometric Identities
The following identities are critical in calculus for simplifying expressions and solving equations involving trigonometric functions.
- Pythagorean Identities:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
\[
1 + \tan^2(\theta) = \sec^2(\theta)
\]
\[
1 + \cot^2(\theta) = \csc^2(\theta)
\] - Reciprocal Identities:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)}
\] - Quotient Identities:
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\]
1.2.5 Trigonometric Values of Special Angles
Certain angles have known sine, cosine, and tangent values that are often used in calculus problems. These include angles like \( 0^\circ \), \( 30^\circ \), \( 45^\circ \), \( 60^\circ \), and \( 90^\circ \), or their radian equivalents \( 0 \), \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
\[
\begin{array}{|c|c|c|c|}
\hline
\theta & \sin(\theta) & \cos(\theta) & \tan(\theta) \\
\hline
0^\circ, \left( 0 \right) & 0 & 1 & 0 \\
\hline
30^\circ, \left( \frac{\pi}{6} \right) & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{3}} \\
\hline
45^\circ, \left( \frac{\pi}{4} \right) & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 \\
\hline
60^\circ, \left( \frac{\pi}{3} \right) & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} \\
\hline
90^\circ, \left( \frac{\pi}{2} \right) & 1 & 0 & \text{undefined} \\
\hline
\end{array}
\]
1.2.6 Graphs of Trigonometric Functions
Trigonometric functions are periodic, meaning they repeat their values at regular intervals. The period of sine and cosine functions is (2\pi), while the period of the tangent function is ( \pi ).
- Graph of Sine Function:
\[
y = \sin(x)
\]
The sine function has amplitude 1, period (2\pi), and oscillates between (-1) and (1). - Graph of Cosine Function:
\[
y = \cos(x)
\]
Like the sine function, the cosine function has amplitude 1 and period \(2\pi\), oscillating between \(-1\) and \(1\). - Graph of Tangent Function:
\[
y = \tan(x)
\]
The tangent function has a period of \( \pi \), and it has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) where \( n \in \mathbb{Z} \).
1.2.7 Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find the angle \( \theta \) given the value of a trigonometric function. The inverse functions are:
- Arcsine:
\[
y = \sin^{-1}(x) \quad \text{means} \quad x = \sin(y) \quad \text{for} \quad -1 \leq x \leq 1 \quad \text{and} \quad -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}
\] - Arccosine:
\[
y = \cos^{-1}(x) \quad \text{means} \quad x = \cos(y) \quad \text{for} \quad -1 \leq x \leq 1 \quad \text{and} \quad 0 \leq y \leq \pi
\] - Arctangent:
\[
y = \tan^{-1}(x) \quad \text{means} \quad x = \tan(y) \quad \text{for} \quad -\infty < x < \infty \quad \text{and} \quad -\frac{\pi}{2} < y < \frac{\pi}{2}
\]
1.2.8 Trigonometric Identities for Calculus
In calculus, certain trigonometric identities frequently appear, especially in the differentiation and integration of trigonometric functions.
- Sum and Difference Identities:
\[
\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)
\]
\[
\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)
\] - Double Angle Identities:
\[
\sin(
2\theta) = 2\sin(\theta)\cos(\theta)
\]
\[
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\]
or equivalently:
\[
\cos(2\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)
\]
These identities are crucial in solving integrals and derivatives involving trigonometric functions.
This concludes the review of trigonometry. Mastery of these concepts will help you navigate calculus problems involving trigonometric functions, particularly as you encounter derivatives and integrals of these functions in later sections.