Estimate differences

Learning Objectives Recall and apply small angle approximations for sine and cosine in radians. Convert small angles from degrees to radians for use in estimations. Select and apply the correct sum-to-product identity to rewrite a trigonometric difference. Combine sum-to-product identities with small angle approximations to estimate the value of a trigonometric difference. Approximate the value of the 'average angle' term using well-known trigonometric values (e.g., for 30°, 45°, 60°). Analyze a problem and execute a multi-step process to find a numerical estimate without a calculator. How do engineers model the tiny vibrations in a bridge or physicists calculate the subtle shift in a light wave? 🌉 They often estimate tiny differences using trigonometry! In this...

TermDefinitionExample Small Angle ApproximationA method used to approximate the values of trigonometric functions for very small angles. For this method to work, the angle must be measured in radians.For a small angle θ = 0.02 radians, sin(θ) ≈ 0.02. The actual value of sin(0.02) is 0.0199986..., which is very close. Radian MeasureThe standard unit of angular measure based on the radius of a circle. Approximations in this topic require angles to be in radians.To convert 2° to radians, we calculate: 2° * (π / 180°) = π/90 radians. Sum-to-Product IdentitiesTrigonometric identities that convert the sum or difference of two trigonometric functions into a product of functions.The difference sin(75°) - sin(15°) can be converted into the product 2cos(45°)sin(30°). Trigonometric DifferenceAn expr...

Sum-to-Product Identity (Sine Difference) sin(A) - sin(B) = 2cos((A+B)/2)sin((A-B)/2) Use this identity to transform the difference of two sine values into a product. This is the first step in our estimation process, as it isolates the small difference angle (A-B)/2. Sum-to-Product Identity (Cosine Difference) cos(A) - cos(B) = -2sin((A+B)/2)sin((A-B)/2) Use this identity to transform the difference of two cosine values into a product. Note the negative sign in front. It also isolates the small difference angle (A-B)/2. Small Angle Approximation for Sine sin(θ) ≈ θ, where θ is a small angle in radians. After using a sum-to-product identity, the term with the small difference angle, sin((A-B)/2), can be approximated by the angle itself, (A-B)/2, provided it's in...