1 / 23

TRIGONOMETRY

TRIGONOMETRY. http://math.la.asu.edu/~tdalesan/mat170/TRIGONOMETRY.ppt. Angles, Arc length, Conversions. Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free

cocheta
Download Presentation

TRIGONOMETRY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. TRIGONOMETRY http://math.la.asu.edu/~tdalesan/mat170/TRIGONOMETRY.ppt

  2. Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and –300° are all coterminal. Degrees to radians: Multiply angle by radians Radians to degrees: Multiply angle by Note: 1 revolution = 360° = 2π radians. Arc length = central angle x radius, or Note: The central angle must be in radian measure.

  3. Right Triangle Trig Definitions B • sin(A) = sine of A = opposite / hypotenuse = a/c • cos(A) = cosine of A = adjacent / hypotenuse = b/c • tan(A) = tangent of A = opposite / adjacent = a/b • csc(A) = cosecant of A = hypotenuse / opposite = c/a • sec(A) = secant of A = hypotenuse / adjacent = c/b • cot(A) = cotangent of A = adjacent / opposite = b/a c a A C b

  4. Special Right Triangles 30° 45° 2 1 60° 45° 1 1

  5. Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Odd functions Reciprocal Identities: Pythagorean Identities:

  6. All Students Take Calculus. Quad I Quad II cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV Quad III

  7. Reference Angles Quad I Quad II θ’ = 180° – θ θ’ = θ θ’ = π – θ θ’ = θ– 180° θ’ = 360° – θ θ’ = θ– π θ’ = 2π – θ Quad III Quad IV

  8. Unit circle • Radius of the circle is 1. • x = cos(θ) • y = sin(θ) • Pythagorean Theorem: • This gives the identity: • Zeros of sin(θ) are where n is an integer. • Zeros of cos(θ) are where n is an integer.

  9. Graphs of sine & cosine • Fundamental period of sine and cosine is 2π. • Domain of sine and cosine is • Range of sine and cosine is [–|A|+D, |A|+D]. • The amplitude of a sine and cosine graph is |A|. • The vertical shift or average value of sine and cosine graph is D. • The period of sine and cosine graph is • The phase shift or horizontal shift is

  10. Sine graphs y = sin(x) y = sin(x) + 3 y = 3sin(x) y = sin(3x) y = sin(x – 3) y = 3sin(3x-9)+3 y = sin(x) y = sin(x/3)

  11. Graphs of cosine y = cos(x) y = cos(x) + 3 y = 3cos(x) y = cos(3x) y = cos(x – 3) y = 3cos(3x – 9) + 3 y = cos(x) y = cos(x/3)

  12. Tangent and cotangent graphs • Fundamental period of tangent and cotangent is π. • Domain of tangent is where n is an integer. • Domain of cotangent where n is an integer. • Range of tangent and cotangent is • The period of tangent or cotangent graph is

  13. Graphs of tangent and cotangent y = tan(x) Vertical asymptotes at y = cot(x) Verrical asymptotes at

  14. Graphs of secant and cosecant y = csc(x) Vertical asymptotes at Range: (–∞, –1] U [1, ∞) y = sin(x) y = sec(x) Vertical asymptotes at Range: (–∞, –1] U [1, ∞) y = cos(x)

  15. Inverse Trigonometric Functions and Trig Equations Domain: [–1, 1] Range: 0 < y < 1, solutions in QI and QII. –1 < y < 0, solutions in QIII and QIV. Domain: Range: Domain: [–1, 1] Range: [0, π] 0 < y < 1, solutions in QI and QIV. –1< y < 0, solutions in QII and QIII. 0 < y < 1, solutions in QI and QIII. –1 < y < 0, solutions in QII and QIV.

  16. Trigonometric IdentitiesSummation & Difference Formulas

  17. Trigonometric IdentitiesDouble Angle Formulas

  18. Trigonometric IdentitiesHalf Angle Formulas The quadrant of determines the sign.

  19. Law of Sines & Law of Cosines Law of sines Law of cosines Use when you have a complete ratio: SSA. Use when you have SAS, SSS.

  20. Vectors • A vector is an object that has a magnitude and a direction. • Given two points P1: and P2: on the plane, a vector v that connects the points from P1 to P2 is v = i + j. • Unit vectors are vectors of length 1. • i is the unit vector in the x direction. • j is the unit vector in the y direction. • A unit vector in the direction of v is v/||v|| • A vector v can be represented in component form by v = vxi + vyj. • The magnitude of v is ||v|| = • Using the angle that the vector makes with x-axis in standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j

  21. Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number). Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j. Dot Product: Multiplication of two vectors. Let v = vxi + vyj, w = wxi + wyj. v·w = vxwx + vywy Example: Let v = 5i + 4j, w = –2i + 3j. v · w = (5)(–2) + (4)(3) = –10 + 12 = 2. Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors. v θ w Two vectors v and w are orthogonal (perpendicular) iff v · w = 0. Addition/subtraction of vectors: Add/subtract same components. Example Let v = 5i + 4j, w = –2i + 3j. v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j. 3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j. ||3v – 2w|| =

  22. Acknowledgements • Unit Circle:http://www.davidhardison.com/math/trig/unit_circle.gif • Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.

More Related