Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
CCSS.MATH.CONTENT.HSF.IF.C.7.A
Graph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.MATH.CONTENT.HSF.IF.C.7.B
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
CCSS.MATH.CONTENT.HSF.IF.C.7.C
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
CCSS.MATH.CONTENT.HSF.IF.C.7.D
(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
CCSS.MATH.CONTENT.HSF.IF.C.7.E
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number.
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Find inverse functions.
CCSS.MATH.CONTENT.HSF.BF.B.4.A
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
CCSS.MATH.CONTENT.HSF.BF.B.4.B
(+) Verify by composition that one function is the inverse of another.
CCSS.MATH.CONTENT.HSF.BF.B.4.C
(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
CCSS.MATH.CONTENT.HSF.BF.B.4.D
(+) Produce an invertible function from a non-invertible function by restricting the domain.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.