Trigonometric Functions: Learn It 3

Trigonometric Identities

Trigonometric identities are the equalities that involve trigonometric functions and are true for any substitution of the variable where the sides of the equation are defined. Mastery of these identities is crucial for solving trigonometric equations, proving other mathematical statements, and is frequently necessary in calculus.

The main trigonometric identities are listed below.

Trigonometric Identities

Reciprocal identities

[latex]\begin{array}{cccc}\tan \theta =\large \frac{\sin \theta}{\cos \theta} & & & \cot \theta =\large \frac{\cos \theta}{\sin \theta} \\ \csc \theta =\large \frac{1}{\sin \theta} & & & \sec \theta =\large \frac{1}{\cos \theta} \end{array}[/latex]
 

 

Pythagorean identities

[latex]\sin^2 \theta +\cos^2 \theta =1\phantom{\rule{2em}{0ex}}1+\tan^2 \theta =\sec^2 \theta \phantom{\rule{2em}{0ex}}1+\cot^2 \theta =\csc^2 \theta[/latex]
 

 

Addition and subtraction formulas

[latex]\sin(\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta[/latex]
[latex]\cos(\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta[/latex]
 

 

Double-angle formulas

[latex]\sin(2\theta)=2\sin \theta \cos \theta[/latex]
[latex]\cos(2\theta)=2\cos^2 \theta -1=1-2\sin^2 \theta =\cos^2 \theta -\sin^2 \theta[/latex]

Understanding and remembering these identities can seem daunting at first. However, with consistent practice and some mnemonic devices, they can become second nature. Below are some study tips:

  • Association: Link each identity with a visual cue or a part of the unit circle. For example, the sine function starts at zero and goes up, just like a sine wave starts at the middle and rises.
  • Mnemonics: Use phrases to remember relationships, like the mnemonic SOHCAHTOA.
  • Repetition: Regularly practice rewriting the identities from memory. Repetition is key to retention.
  • Flashcards: Create a set of flashcards with each identity on one side and its name or a key hint on the other.
  • Group Study: Discuss and solve problems with peers; explaining concepts to others can reinforce your memory.

When facing a trigonometric identity, verification is key to ensuring the identity holds true for all permissible values of the variable. The process is a methodical one, where you manipulate one side of the equation until it matches the other.

How to: Given a Trigonometric Identity, Verify that it is True.

  1. Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
  2. Look for opportunities to factor expressions, square a binomial, or add fractions.
  3. Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
  4. If these steps do not yield the desired result, try converting all terms to sines and cosines.

Verify [latex]\tan{\theta} \cos{\theta} = \sin{\theta}[/latex].

Prove the trigonometric identity [latex]1+\tan^2 \theta =\sec^2 \theta[/latex].

Algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

The equation [latex](\sin x+1)(\sin x−1)=0[/latex] resembles the equation [latex](x+1)(x−1)=0[/latex], which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. 

How to: Given a Trigonometric Equation, Solve using Algebra.

  1. Spot Patterns: Look for familiar algebraic cues in the equation, like notable identities or factors.
  2. Substitute Variables: Temporarily replace trig expressions with a single variable to simplify the equation.
  3. Solve Algebraically: Treat the simplified equation as you would a standard algebraic one.
  4. Back-Substitute: Once you’ve solved for the temporary variable, revert to the original trigonometric terms.
  5. Find Angles: Use inverse functions to solve for the angle, considering the function’s period and domain.
  6. Verify Solutions: Always check that your solutions satisfy the initial equation.

Write the following trigonometric expression as an algebraic expression:

[latex]2 \cos^2{\theta} + \cos{\theta} -1[/latex]

 

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