Lecture b4 Exam 1

  1. Key Concepts

    • Understanding key angles and their cosine values.

    • The unit circle is essential for determining angles based on cosine values.

    • Problem-solving involves finding angles where trigonometric functions equal specific values and recognizing quadrant reflections.

  2. Definitions

    • Trigonometric Functions: Functions related to the angles of a triangle, specifically sine, cosine, and tangent.

    • Coterminal Angles: Angles that share the same terminal side, differing by full rotations (360 degrees).

    • Quadrants: The four sections of a coordinate plane, each defined by the signs of x (horizontal) and y (vertical) coordinates.

  3. Formulas Introduced

    • Inverse sine function: sin^{-1}(x) to find angles.

    • Coterminal angle formula: Original angle + 360k (where k is any integer).

    • Pythagorean theorem: a^2 + b^2 = c^2 for triangle calculations.

  4. Step-by-Step Worked Examples from Class
    Example 1: Finding Sine
    Step 1: Identify the angle for 270 degrees.
    Step 2: Compute: sin(270) = -1.

    Example 2: Coterminal Angle
    Step 1: Given angle = -26.744 degrees.
    Step 2: Add 360 degrees.
    Step 3: Calculate: -26.744 + 360 = 333.256 degrees.

    Example 3: Trigonometric Ratios Using a Triangle
    Step 1: For cosine = -1/4, draw a right triangle.
    Step 2: Identify opposite and adjacent sides relative to the angle.
    Step 3: Use ratios to resolve other angles and sides based on the identified triangle.

  5. Professor Warnings / Exam Hints

    • Always check the accuracy of calculations related to degree contexts.

    • Memorize the unit circle, as it provides necessary context for angle calculations.

    • Pay attention to quadrant reflections when determining angles from trigonometric functions.

  6. Common Mistakes Mentioned

    • Forgetting to account for the correct signs of sine, cosine, and tangent in different quadrants.

    • Miscalculating coterminal angles by not adding or subtracting 360 degrees appropriately.

  7. Questions Students Asked

    • What is the best way to remember the unit circle?

    • How do I accurately identify which quadrant an angle is in?

    • Are there shortcuts for solving trigonometric identities?