Trigonometry Through Wayfinding and Navigation Across the Pacific

HSA.CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.(Section 2.3)

HSF.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Section 2.1, Section 2.2)

HSF.IF.C.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (Section 2.1, Section 2.2)

HSF.BF.A.1: Write a function that describes a relationship between two quantities. (Section 2.3)

HSF.BF.B.3: Identify the effect on the graph of replacing \(f(x)\) by \(f(x) + k\text{,}\) \(k f(x)\text{,}\) \(f(kx)\text{,}\) and \(f(x + k)\) for specific values of \(k\) (both positive and negative); find the value of \(k\) given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (Subsection 2.1.4, Subsection 2.2.9)

HSF.TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (Definition 1.2.18)

HSF.TF.A.2: 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. (Subsection 1.3.2)

HSF.TF.A.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for \(\pi/3\text{,}\) \(\pi/4\) and \(\pi/6\text{,}\) and use the unit circle to express the values of sine, cosine, and tangent for \(x\text{,}\) \(\pi+x\text{,}\) and \(2\pi-x\) in terms of their values for \(x\text{,}\) where \(x\) is any real number. (Subsection 1.4.2, Subsection 1.5.3)

HSF.TF.A.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions. (Subsection 1.5.4, Subsection 1.5.7)

HSF.TF.B.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. (Subsection 2.1.4, Section 2.3)

HSF.TF.B.6: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (Subsection 2.4.1)

HSF.TF.B.7: 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. (Exercise 2.1.5.46, Exercise Group 2.1.5.47–50, Exercise Group 2.2.10.37–42, Exercise Group 2.2.10.43–52, Example 2.4.18, Example 2.4.19, Example 2.4.17)

HSF.TF.C.8: Prove the Pythagorean identity \(\sin^2\theta+\cos^2\theta = 1\) and use it to find \(\sin\theta\text{,}\) \(\cos\theta\text{,}\) or \(\tan\theta\) given \(\sin\theta\text{,}\) \(\cos\theta\text{,}\) or \(\tan\theta\) and the quadrant of the angle. (Definition 1.5.20)

HSF.TF.C.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (Proof 3.2.1.1, Example 3.2.10, Proof 3.2.3.1)

HSG.SRT.C.6: 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. (Definition 1.4.1)

HSG.SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles. (Definition 1.4.7, Remark 1.4.9)

HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (Subsection 1.4.6)

HSG.SRT.D.9: Derive the formula \(\mbox{Area}=\frac{1}{2}ab\sin(C)\) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (Definition 4.1.11)

HSG.SRT.D.9: Prove the Laws of Sines and Cosines and use them to solve problems. (Subsection 4.1.1, Subsection 4.2.1)

HSG.SRT.D.9: 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).

HSG.C.B.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (Definition 1.2.18, Theorem 1.2.25, Definition 1.2.28)