🔺Trigonometry Unit 7 – Trigonometric Identities

Key Concepts and Definitions

• Trigonometric identities are equations that are true for all values of the variable for which both sides of the equation are defined
• Identities are used to simplify trigonometric expressions, solve trigonometric equations, and prove other identities
• Trigonometric functions include sine ($\sin$), cosine ($\cos$), tangent ($\tan$), cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$)
• Trigonometric functions are defined in terms of the sides of a right triangle (opposite, adjacent, and hypotenuse)
• Angle measure can be expressed in degrees or radians, where $\pi$ radians equals 180 degrees
• Periodicity refers to the repeating nature of trigonometric functions, with sine and cosine having a period of $2\pi$ and tangent having a period of $\pi$
• Even and odd functions describe the symmetry of trigonometric functions, with cosine being an even function and sine and tangent being odd functions

Fundamental Trigonometric Identities

• Reciprocal identities relate trigonometric functions to their reciprocals, such as $\sin\theta = \frac{1}{\csc\theta}$ and $\cos\theta = \frac{1}{\sec\theta}$
  • Other reciprocal identities include $\tan\theta = \frac{1}{\cot\theta}$, $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, and $\cot\theta = \frac{1}{\tan\theta}$
• Quotient identities express trigonometric functions as ratios of other functions, like $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$
• Negative angle identities relate the values of trigonometric functions for negative angles to their positive counterparts
  • For example, $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$, and $\tan(-\theta) = -\tan\theta$
• Cofunction identities relate trigonometric functions of complementary angles, such as $\sin(\frac{\pi}{2} - \theta) = \cos\theta$ and $\cos(\frac{\pi}{2} - \theta) = \sin\theta$
• Odd and even identities describe the behavior of trigonometric functions under reflection, with $\sin(-\theta) = -\sin\theta$ (odd) and $\cos(-\theta) = \cos\theta$ (even)

Pythagorean Identities

• Pythagorean identities are derived from the Pythagorean theorem and relate the squares of trigonometric functions
• The fundamental Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$
  • This identity can be derived by dividing the Pythagorean theorem equation by the square of the hypotenuse
• Other Pythagorean identities include $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$
• These identities are useful for simplifying trigonometric expressions and solving equations
  • For example, to simplify $\sin^2\theta - \cos^2\theta$, you can use the Pythagorean identity to rewrite it as $1 - 2\cos^2\theta$
• Pythagorean identities can also be used to find the values of other trigonometric functions when one function value is known

Sum and Difference Identities

• Sum and difference identities express the sine, cosine, or tangent of the sum or difference of two angles in terms of the sines and cosines of the individual angles
• The sum identities for sine and cosine are $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$
• The difference identities for sine and cosine are $\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$ and $\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$
• The tangent sum and difference identities are $\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$ and $\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}$
  • These identities can be derived using the sum and difference identities for sine and cosine, along with the quotient identity for tangent
• Sum and difference identities are useful for solving trigonometric equations, simplifying expressions, and proving other identities

Double Angle and Half Angle Formulas

• Double angle formulas express the sine, cosine, or tangent of twice an angle in terms of the trigonometric functions of the original angle
• The double angle formulas for sine and cosine are $\sin(2\theta) = 2\sin\theta\cos\theta$ and $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
  • An alternative formula for cosine is $\cos(2\theta) = 2\cos^2\theta - 1$, which can be derived using the Pythagorean identity
• The double angle formula for tangent is $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$
• Half angle formulas express the sine, cosine, or tangent of half an angle in terms of the trigonometric functions of the original angle
  • For example, $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$ and $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$
• Double angle and half angle formulas are useful for solving trigonometric equations and simplifying expressions

Practical Applications

• Trigonometric identities have numerous applications in various fields, such as physics, engineering, and navigation
• In physics, trigonometric identities are used to analyze wave phenomena, such as sound and light waves, and to solve problems involving vectors and forces
  • For example, the sum and difference identities can be used to analyze the interference patterns created by multiple waves
• In engineering, trigonometric identities are used to design and analyze structures, such as bridges and buildings, and to study the motion of machines and mechanisms
  • The Pythagorean identities can be used to calculate the forces acting on a structure or the torque generated by a rotating machine
• In navigation, trigonometric identities are used to calculate distances, angles, and positions on the Earth's surface
  • The spherical law of cosines, which is derived from the Pythagorean identity, is used to calculate the distance between two points on a sphere given their latitudes and longitudes
• Trigonometric identities are also used in computer graphics to rotate and transform objects in 2D and 3D space
  • The double angle formulas are used to efficiently calculate the sine and cosine of angles that are multiples of a base angle

Common Mistakes and How to Avoid Them

• One common mistake is confusing trigonometric identities with equations
  • Remember that identities are true for all values of the variable, while equations are only true for specific values
• Another mistake is not properly applying the domain restrictions when using identities
  • For example, the Pythagorean identity $\tan^2\theta + 1 = \sec^2\theta$ is only valid for angles where $\cos\theta \neq 0$
• Forgetting to simplify expressions or leaving them in an unsimplified form can lead to errors
  • Always simplify expressions as much as possible using the appropriate identities
• Misapplying identities or using them in the wrong context can lead to incorrect results
  • Make sure to choose the appropriate identity for the problem at hand and use it correctly
• Not checking the final answer for reasonableness can result in accepting incorrect solutions
  • Always double-check your work and verify that the final answer makes sense in the context of the problem
• Memorizing identities without understanding their derivations or applications can hinder problem-solving skills
  • Focus on understanding the concepts behind the identities and practice applying them to various problems

Practice Problems and Solutions

1. Simplify the expression $\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta$.

   • Solution: $\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta = (\sin\theta - \cos\theta)^2 + \sin\theta\cos\theta = (\sin\theta - \cos\theta)^2 + \frac{1}{2}\sin(2\theta) = 1 - \sin(2\theta) + \frac{1}{2}\sin(2\theta) = 1 - \frac{1}{2}\sin(2\theta)$

2. Prove the identity $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$.

   • Solution: $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = \frac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)} = \frac{1 - \cos^2\theta + 1 + 2\cos\theta + \cos^2\theta}{\sin\theta(1 + \cos\theta)} = \frac{2(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)} = \frac{2}{\sin\theta} = 2\csc\theta$

3. If $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, find the values of the other five trigonometric functions.

   • Solution: Given $\sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, we can find $\cos\theta$ using the Pythagorean identity: $\cos^2\theta = 1 - \sin^2\theta = 1 - (\frac{3}{5})^2 = \frac{16}{25}$. Since $\theta$ is in Quadrant II, $\cos\theta$ is negative, so $\cos\theta = -\frac{4}{5}$. Using the quotient identities, we can find $\tan\theta = \frac{\sin\theta}{\cos\theta} = -\frac{3}{4}$, $\csc\theta = \frac{1}{\sin\theta} = \frac{5}{3}$, $\sec\theta = \frac{1}{\cos\theta} = -\frac{5}{4}$, and $\cot\theta = \frac{1}{\tan\theta} = -\frac{4}{3}$.

4. Simplify the expression $\cos(3\theta)$ using the triple angle formula.

   • Solution: Using the triple angle formula for cosine, $\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$. No further simplification is possible without knowing the value of $\cos\theta$.

5. Solve the equation $2\sin^2\theta - \cos\theta = 1$ for $\theta$ in the interval $[0, 2\pi]$.

   • Solution: Substitute $\cos^2\theta = 1 - \sin^2\theta$ into the equation: $2\sin^2\theta - \sqrt{1 - \sin^2\theta} = 1$. Square both sides to eliminate the square root: $4\sin^4\theta - 4\sin^2\theta + 1 = 1 - \sin^2\theta$. Simplify and factor: $4\sin^4\theta - 3\sin^2\theta = 0$, $\sin^2\theta(4\sin^2\theta - 3) = 0$. Solve for $\sin^2\theta$: $\sin^2\theta = 0$ or $\sin^2\theta = \frac{3}{4}$. Take the square root and consider the sign: $\sin\theta = 0$, $\sin\theta = \pm\frac{\sqrt{3}}{2}$. In the interval $[0, 2\pi]$, the solutions are $\theta = 0$, $\theta = \frac{\pi}{3}$, $\theta = \frac{2\pi}{3}$, and $\theta = \pi$.