Solve inequalities using estimation

Learning Objectives Estimate the values of trigonometric functions for key angles on the unit circle. Apply fundamental trigonometric identities to simplify inequalities before estimation. Determine the intervals where a trigonometric inequality holds true by estimating function values at test points. Solve inequalities of the form sin(x) > c, cos(x) < c, or tan(x) > c over a given interval using estimation. Compare the magnitudes of two different trigonometric functions (e.g., sin(x) vs. cos(x)) within a specific quadrant by estimation. Verify potential solutions to trigonometric inequalities by substituting estimated values. How do game developers make a character's line of sight work realistically without calculating every single angle in real-time? 🤔 They...

TermDefinitionExample Unit Circle EstimationUsing the (x, y) coordinates on the unit circle to approximate the values of cos(θ) and sin(θ) respectively. The x-coordinate represents cosine, and the y-coordinate represents sine.For an angle slightly less than π/2, like 2π/5, we can estimate that its y-coordinate (sine) is close to 1 and its x-coordinate (cosine) is small and positive. PeriodicityThe property of trigonometric functions to repeat their values at regular intervals. The period of sin(x) and cos(x) is 2π, while for tan(x) it is π.If we know sin(x) > 0.5 for x in (π/6, 5π/6), then due to periodicity, it's also true for x in (π/6 + 2kπ, 5π/6 + 2kπ) for any integer k. Monotonicity on an IntervalThe behavior of a function being consistently increasing or decreasing over a sp...

Sine and Cosine Bounds -1 ≤ sin(x) ≤ 1 and -1 ≤ cos(x) ≤ 1 The output of the sine and cosine functions is always between -1 and 1, inclusive. This provides a quick check for impossible inequalities, such as sin(x) > 2. Pythagorean Identity sin²(x) + cos²(x) = 1 Use this to relate sine and cosine. If an inequality contains both, you can often substitute one to simplify the problem into a single function before estimating. Tangent Identity tan(x) = sin(x) / cos(x) Use this to understand the behavior of the tangent function based on the signs and relative sizes of sine and cosine. For example, where |sin(x)| > |cos(x)|, we know that |tan(x)| > 1.