Unit 9 Chapter 5 Trigonometry Review Notes

Linear Combinations of Sinusoids * General property for transforming a linear combination of sinusoids with equal periods: bcos⁡(x)+csin⁡(x)=Acos⁡(x−D)b\cos(x) + c\sin(x) = A\cos(x - D) * Magnitude formula: A=b2+c2A = \sqrt{b^2 + c^2} * Phase displacement formula: D=arctanc b sinD = \text{arctanc b sin} * Example transformations: y=−3cos⁡(θ)+5sin⁡(θ)y = -3\cos(\theta) + 5\sin(\theta) and y=6cos⁡(x)+7sin⁡(x)y = 6\cos(x) + 7\sin(x) where period and amplitude determine the sinusoid's form. # Trigonometric Identities and Properties * Half-Angle Identities: sin⁡(12x)=±12(1−cos⁡(x))\sin(\frac{1}{2}x) = \pm\sqrt{\frac{1}{2}(1 - \cos(x))} and cos⁡(12x)=±12(1+cos⁡(x))\cos(\frac{1}{2}x) = \pm\sqrt{\frac{1}{2}(1 + \cos(x))} * Double Angle Identities: sin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x) and cos⁡(2x)=1−2sin⁡2(12x)\cos(2x) = 1 - 2\sin^2(\frac{1}{2}x) * Product-to-Sum Identities: 2cos⁡(A)cos⁡(B)=cos⁡(A+B)+cos⁡(A−B)2\cos(A)\cos(B) = \cos(A + B) + \cos(A - B) and 2sin⁡(A)sin⁡(B)=−cos⁡(A+B)+cos⁡(A−B)2\sin(A)\sin(B) = -\cos(A + B) + \cos(A - B) * Sum-to-Product Identities: sin⁡(x)−sin⁡(y)=2cos⁡(12(x+y))sin⁡(12(x−y))\sin(x) - \sin(y) = 2\cos(\frac{1}{2}(x + y))\sin(\frac{1}{2}(x - y)) and cos⁡(x)−cos⁡(y)=−2sin⁡(12(x+y))sin⁡(12(x−y))\cos(x) - \cos(y) = -2\sin(\frac{1}{2}(x + y))\sin(\frac{1}{2}(x - y)) * Power Reduction Formulas: sin⁡2(x)=12(1−cos⁡(2x))\sin^2(x) = \frac{1}{2}(1 - \cos(2x)) and cos⁡2(x)=12(1+cos⁡(2x))\cos^2(x) = \frac{1}{2}(1 + \cos(2x)) # Fundamental Reciprocal and Angle Properties * Reciprocal Identities: cot⁡(x)=1tan⁡(x)\cot(x) = \frac{1}{\tan(x)} * Quotient Identities: tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)} * Pythagorean Identity: cos⁡2(x)+sin⁡2(x)=1\cos^2(x) + \sin^2(x) = 1 * Odd/Even Properties: sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x) (odd function) and cos⁡(−x)=cos⁡(x)\cos(-x) = \cos(x) (even function) * Cofunction Properties: sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta) = \cos(90^\circ - \theta) such as cos⁡(13∘)=sin⁡(77∘)\cos(13^\circ) = \sin(77^\circ) # Analytical Applications and Solutions * Composite Arguments: Simplify expressions like cos⁡(x)cos⁡(71∘)−sin⁡(x)sin⁡(71∘)=0.53\cos(x)\cos(71^\circ) - \sin(x)\sin(71^\circ) = 0.53 and cos⁡(x)cos⁡(25∘)−sin⁡(x)sin⁡(25∘)=0.9\cos(x)\cos(25^\circ) - \sin(x)\sin(25^\circ) = 0.9 into single sinusoids to solve for xx * Exact Values: Calculating values for 2x2x or 12x\frac{1}{2}x given conditions like cos⁡(x)=0.34\cos(x) = 0.34 or cos⁡(A)=−37\cos(A) = -\frac{3}{7} where A∈[90∘,180∘]A \in [90^\circ, 180^\circ] * Graphing Relations: Transforming y=cos⁡(x−2)y = \cos(x - 2) into a linear combination using cos⁡(2)≈−0.42\cos(2) \approx -0.42 and sin⁡(2)≈0.91\sin(2) \approx 0.91 * Algebraic Proofs: Converting product forms like y=4cos⁡(x)cos⁡(11x)y = 4\cos(x)\cos(11x) into sums of sinusoids for numerical verification

Linear Combinations of Sinusoids * General property for transforming a linear combination of sinusoids with equal periods: bcos⁡(x)+csin⁡(x)=Acos⁡(x−D)b\cos(x) + c\sin(x) = A\cos(x - D) * Magnitude formula: A=b2+c2A = \sqrt{b^2 + c^2} * Phase displacement formula: D=arctanc b sinD = \text{arctanc b sin} * Example transformations: y=−3cos⁡(θ)+5sin⁡(θ)y = -3\cos(\theta) + 5\sin(\theta) and y=6cos⁡(x)+7sin⁡(x)y = 6\cos(x) + 7\sin(x) where period and amplitude determine the sinusoid's form. # Trigonometric Identities and Properties * Half-Angle Identities: sin⁡(12x)=±12(1−cos⁡(x))\sin(\frac{1}{2}x) = \pm\sqrt{\frac{1}{2}(1 - \cos(x))} and cos⁡(12x)=±12(1+cos⁡(x))\cos(\frac{1}{2}x) = \pm\sqrt{\frac{1}{2}(1 + \cos(x))} * Double Angle Identities: sin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x) and cos⁡(2x)=1−2sin⁡2(12x)\cos(2x) = 1 - 2\sin^2(\frac{1}{2}x) * Product-to-Sum Identities: 2cos⁡(A)cos⁡(B)=cos⁡(A+B)+cos⁡(A−B)2\cos(A)\cos(B) = \cos(A + B) + \cos(A - B) and 2sin⁡(A)sin⁡(B)=−cos⁡(A+B)+cos⁡(A−B)2\sin(A)\sin(B) = -\cos(A + B) + \cos(A - B) * Sum-to-Product Identities: sin⁡(x)−sin⁡(y)=2cos⁡(12(x+y))sin⁡(12(x−y))\sin(x) - \sin(y) = 2\cos(\frac{1}{2}(x + y))\sin(\frac{1}{2}(x - y)) and cos⁡(x)−cos⁡(y)=−2sin⁡(12(x+y))sin⁡(12(x−y))\cos(x) - \cos(y) = -2\sin(\frac{1}{2}(x + y))\sin(\frac{1}{2}(x - y)) * Power Reduction Formulas: sin⁡2(x)=12(1−cos⁡(2x))\sin^2(x) = \frac{1}{2}(1 - \cos(2x)) and cos⁡2(x)=12(1+cos⁡(2x))\cos^2(x) = \frac{1}{2}(1 + \cos(2x)) # Fundamental Reciprocal and Angle Properties * Reciprocal Identities: cot⁡(x)=1tan⁡(x)\cot(x) = \frac{1}{\tan(x)} * Quotient Identities: tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)} * Pythagorean Identity: cos⁡2(x)+sin⁡2(x)=1\cos^2(x) + \sin^2(x) = 1 * Odd/Even Properties: sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x) (odd function) and cos⁡(−x)=cos⁡(x)\cos(-x) = \cos(x) (even function) * Cofunction Properties: sin⁡(θ)=cos⁡(90∘−θ)\sin(\theta) = \cos(90^\circ - \theta) such as cos⁡(13∘)=sin⁡(77∘)\cos(13^\circ) = \sin(77^\circ) # Analytical Applications and Solutions * Composite Arguments: Simplify expressions like cos⁡(x)cos⁡(71∘)−sin⁡(x)sin⁡(71∘)=0.53\cos(x)\cos(71^\circ) - \sin(x)\sin(71^\circ) = 0.53 and cos⁡(x)cos⁡(25∘)−sin⁡(x)sin⁡(25∘)=0.9\cos(x)\cos(25^\circ) - \sin(x)\sin(25^\circ) = 0.9 into single sinusoids to solve for xx * Exact Values: Calculating values for 2x2x or 12x\frac{1}{2}x given conditions like cos⁡(x)=0.34\cos(x) = 0.34 or cos⁡(A)=−37\cos(A) = -\frac{3}{7} where A∈[90∘,180∘]A \in [90^\circ, 180^\circ] * Graphing Relations: Transforming y=cos⁡(x−2)y = \cos(x - 2) into a linear combination using cos⁡(2)≈−0.42\cos(2) \approx -0.42 and sin⁡(2)≈0.91\sin(2) \approx 0.91 * Algebraic Proofs: Converting product forms like y=4cos⁡(x)cos⁡(11x)y = 4\cos(x)\cos(11x) into sums of sinusoids for numerical verification