(90) Math12550 Exam 3 Review Part 1 of 6: Trigonometric Functions

Trigonometric Functions Overview

• Understanding trigonometric functions is essential for solving various problems in trigonometry.

• Familiarity with radian and degree notation is critical.

• The unit circle is a pivotal tool in finding trigonometric values.

Angle: θ = π/3

• Conversion: π/3 radians = 60 degrees.

• Unit Circle Point: For θ = π/3, the corresponding unit circle point is (1/2, √3/2).

Calculating the Six Trigonometric Functions

• Sine (sin θ): - y-coordinate of the unit circle point.

  • sin(π/3) = √3/2

• Cosine (cos θ): - x-coordinate of the unit circle point.

  • cos(π/3) = 1/2

• Tangent (tan θ): - Ratio of sine to cosine (y/x).

  • tan(π/3) = (√3/2) / (1/2) = √3

• Cosecant (csc θ): - Reciprocal of sine.

  • csc(π/3) = 1/sin(π/3) = 2/√3 (can rationalize if necessary).

• Secant (sec θ): - Reciprocal of cosine.

  • sec(π/3) = 1/cos(π/3) = 2.

• Cotangent (cot θ): - Reciprocal of tangent.

  • cot(π/3) = 1/tan(π/3) = 1/√3 (can rationalize if necessary).

Special Triangles

• Knowing the special 30-60-90 triangle helps derive trigonometric functions.

  • For 60 degrees (π/3), the triangle sides are (1, 2, √3).

  • Sine (sin): Opposite / Hypotenuse = √3/2.

  • Cosine (cos): Adjacent / Hypotenuse = 1/2.

Angle: θ with Sine -4/5 in Quadrant III

• **Quadrant III Characteristics:

  • Sine: Negative

  • Cosine: Negative

  • Tangent: Positive**

• Given: sin θ = -4/5.

Constructing the Right Triangle

• Ignore the sign for calculation initially:

  • Opposite side = 4, Hypotenuse = 5.

  • Using Pythagorean theorem: adjacent side = 3.

Calculating Trigonometric Functions

• Cosine (cos θ): - Adjacent / Hypotenuse = -3/5.

• Tangent (tan θ): - Opposite / Adjacent = 4/3 (positive in Quadrant III).

• Other Functions Calculation:

  • Cosecant (csc θ): 1/sin θ = -5/4.

  • Secant (sec θ): 1/cos θ = -5/3.

  • Cotangent (cot θ): 1/tan θ = 3/4.

Importance of Drawing Triangles

• Draw a triangle for clarity and determine the relationships between the sides.

• Remembering signs based on quadrant helps ascertain the correct values of trigonometric functions.