Precalculus Study Guide: Exponential Functions, Trigonometric Identities, and Laws

Exponential Functions and Applications

Compound Interest

Compound interest is a fundamental concept in finance, describing how an initial principal grows over time when interest is added periodically. The formulas below are used to calculate the future value of an investment.

• Compound Interest Formula (n times per year):

• $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compound Interest Formula:

• $A = P e^{rt}$

• Definitions:

  • P: Principal (initial amount)

  • r: Annual interest rate (as a decimal)

  • n: Number of compounding periods per year

  • t: Time in years

  • A: Amount after t years

• Example: If $P = 1000$, $r = 0.05$, $n = 4$, $t = 3$, then $A = 1000 \left(1 + \frac{0.05}{4}\right)^{12}$.

Exponential Growth and Decay

Exponential growth and decay describe processes where quantities increase or decrease at rates proportional to their current value. These are common in population dynamics, radioactive decay, and finance.

• General Formula:

• $P(t) = P_0 e^{kt}$

• $A(t) = A_0 e^{kt}$

• Definitions:

  • P_0 or A_0: Initial amount

  • k: Growth (k > 0) or decay (k < 0) constant

  • t: Time

• Example: If a population starts at 500 and grows at a rate $k = 0.02$, then after 10 years: $P(10) = 500 e^{0.2}$.

Trigonometric Identities and Formulas

Sum and Difference Formulas

Sum and difference formulas allow the calculation of trigonometric functions for sums or differences of angles. These are essential for simplifying expressions and solving equations.

• Sine:

• $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

• $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$

• Cosine:

• $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

• $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

• Tangent:

• $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

• $\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

• Example: Find $\sin(75^\circ)$ using $\sin(45^\circ + 30^\circ)$.

Double Angle and Power Reducing Formulas

Double angle formulas express trigonometric functions of twice an angle in terms of the original angle. Power reducing formulas rewrite powers of trigonometric functions in terms of first powers.

• Double Angle for Tangent:

• $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

• Power Reducing for Tangent:

• $\tan^2 \theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$

• Example: Simplify $\tan^2(\theta)$ using the power reducing formula.

Half Angle Formulas

Half angle formulas are used to find the sine, cosine, or tangent of half an angle. They are useful in integration and solving trigonometric equations.

• $\cos \left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + \cos \alpha}{2}}$

• $\sin \left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos \alpha}{2}}$

• $\tan \left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}$

• $\tan \left(\frac{\alpha}{2}\right) = \frac{\sin \alpha}{1 + \cos \alpha}$

• $\tan \left(\frac{\alpha}{2}\right) = \frac{1 - \cos \alpha}{\sin \alpha}$

• Example: Find $\sin(15^\circ)$ using the half angle formula.

Product to Sum Formulas

Product to sum formulas convert products of sines and cosines into sums, which are useful in simplifying trigonometric expressions and solving integrals.

• $\sin \alpha \cos \beta = \frac{1}{2} [\sin(\alpha + \beta) + \sin(\alpha - \beta)]$

• $\cos \alpha \sin \beta = \frac{1}{2} [\sin(\alpha + \beta) - \sin(\alpha - \beta)]$

• $\sin \alpha \sin \beta = \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)]$

• $\cos \alpha \cos \beta = \frac{1}{2} [\cos(\alpha - \beta) + \cos(\alpha + \beta)]$

• Example: Simplify $\sin 3x \cos 2x$ using product to sum.

Sum to Product Formulas

Sum to product formulas convert sums or differences of sines and cosines into products. These are useful for solving trigonometric equations.

• $\sin \alpha + \sin \beta = 2 \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right)$

• $\cos \alpha + \cos \beta = 2 \cos \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right)$

• $\sin \alpha - \sin \beta = 2 \sin \left(\frac{\alpha - \beta}{2}\right) \cos \left(\frac{\alpha + \beta}{2}\right)$

• $\cos \alpha - \cos \beta = -2 \sin \left(\frac{\alpha + \beta}{2}\right) \sin \left(\frac{\alpha - \beta}{2}\right)$

• Example: Express $\sin 70^\circ + \sin 50^\circ$ as a product.

Trigonometric Laws

Law of Sines

The Law of Sines relates the sides and angles of a triangle, allowing for the solution of unknowns in non-right triangles.

• $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

• Definitions:

  • A, B, C: Angles of the triangle

  • a, b, c: Sides opposite those angles

• Example: Given $A = 30^\circ$, $a = 10$, $B = 45^\circ$, find $b$.

Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for any triangle, relating the lengths of sides to the cosine of an included angle.

• $a^2 = b^2 + c^2 - 2bc \cos A$

• $b^2 = a^2 + c^2 - 2ac \cos B$

• $c^2 = a^2 + b^2 - 2ab \cos C$

• Definitions:

  • A, B, C: Angles of the triangle

  • a, b, c: Sides opposite those angles

• Example: Given $a = 7$, $b = 5$, $C = 60^\circ$, find $c$.

The Unit Circle

Definition and Properties

The unit circle is a circle of radius 1 centered at the origin in the coordinate plane. It is fundamental in trigonometry for defining the sine and cosine of angles.

• Equation: $x^2 + y^2 = 1$

• Key Points:

  • For an angle $\theta$, the coordinates $(x, y)$ on the unit circle are $(\cos \theta, \sin \theta)$.

  • Common angles: $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, etc.

  • Used to derive values of trigonometric functions and their signs in different quadrants.

• Example: At $\theta = 45^\circ$, the coordinates are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Additional info: Academic context and examples were added to expand brief formulas into a comprehensive study guide.