Trigonometry

Study of triangles and trigonometric functions including sine, cosine, and tangent.

Understanding Angles and TrianglesSine, Cosine, and Tangent FunctionsThe Unit Circle and Radian Measure
Trigonometric Identities and FormulasSolving TrianglesGraphs of Trigonometric Functions
Measuring Heights and DistancesTrigonometry in Engineering and DesignNavigation and GPS
Memorizing Trigonometric ValuesSolving Word Problems with DiagramsChecking Your Work with Inverse Functions
Understanding Angles...Sine, Cosine, and Ta...The Unit Circle and ...Trigonometric Identi...Solving TrianglesGraphs of Trigonomet...Measuring Heights an...Trigonometry in Engi...Navigation and GPSMemorizing Trigonome...Solving Word Problem...Checking Your Work w...
Advanced Topics

Trigonometric Identities and Formulas

Powerful Shortcuts: Trigonometric Identities

Trigonometric identities are like secret codes that simplify tricky calculations. They show how different trig functions relate to each other.

Key Identities

  • Pythagorean: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • Reciprocal: \( \sec(\theta) = \frac{1}{\cos(\theta)} \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
  • Tangent as Sine over Cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

These formulas let you rewrite and solve equations faster.

Key Formula

\[\sin^2(\theta) + \cos^2(\theta) = 1\]

Examples

  • If \( \sin(\theta) = 0.6 \), then \( \cos(\theta) = 0.8 \) since \( 0.6^2 + 0.8^2 = 1 \).

  • Knowing \( \tan(45^\circ) = 1 \), you can check with \( \frac{\sin(45^\circ)}{\cos(45^\circ)} = 1 \).

In a Nutshell

Identities help simplify expressions and solve equations in trigonometry.

Key Terms

Identity
An equation that holds true for all values of the variables.
Reciprocal
The flipped version of a number or function (e.g., 1/x is the reciprocal of x).
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Solving Triangles