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Trigonometry

Trigonometry. Trigonometry is anything using sin, cos or tan. Trigonometry - Radians. What is a radian ?. r. r. One radian is the angle of a circle which has an arc length equal to the radius. There are 2 π radians in 360°. 1 c. r. Trig – Radians to degrees. Converting:.

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Trigonometry

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  1. Trigonometry Trigonometry is anything using sin, cos or tan Maths revision course by Miriam Hanks

  2. Trigonometry - Radians What is a radian ? r r One radian is the angle of a circle which has an arc length equal to the radius. There are 2π radians in 360° 1c r Maths revision course by Miriam Hanks

  3. Trig – Radians to degrees Converting: radians degrees degrees radians Maths revision course by Miriam Hanks

  4. Trigonometry – Exact values • Your exact values need to be learned: Maths revision course by Miriam Hanks

  5. Trigonometry – Graphs y = sin x Maths revision course by Miriam Hanks

  6. Trigonometry – Graphs y = cos x Maths revision course by Miriam Hanks

  7. Trigonometry – Graphs y = tan x Maths revision course by Miriam Hanks

  8. Trigonometry – CAST When do you use CAST? Use CAST quadrants whenever you press sin-1, or cos-1 or tan-1 on your calculator. 180 - x x S A T C 180 + x 360 - x Maths revision course by Miriam Hanks

  9. Trigonometry – formulae Remember: sin2 x means (sinx)2 sin 2x = 2 sin x cos x cos 2x = cos2 x – sin2 x cos 2x = 2 cos2 x – 1 cos 2x = 1 – 2sin2 x All these formulae are on the formula sheet Maths revision course by Miriam Hanks

  10. Trigonometry – formulae sin (p ± q) = sin p cos q ± cos p sin q cos (p ± q) = cos p cos q + sin p sin q These are also on the formula sheet Maths revision course by Miriam Hanks

  11. Trigonometry – formulae These 2 are NOTon the formula sheet Maths revision course by Miriam Hanks

  12. Solving trig equations • If there are “double angles and single angles”, get rid of the double angles by using the formulae on the sheet for sin 2A or cos 2A. • eg 3sinx – 1 = cos 2x single angle double angle 3sinx – 1 = 1 – 2sin2x Maths revision course by Miriam Hanks

  13. Solving trig equations • If there is a squared term, then put everything onto one side, and factorise. • 3sinx – 1 = 1 – 2sin2x squared term 2sin2x + 3sinx – 2 = 0 (2sinx – 1)(sinx + 2)= 0 sin x = ½ or sinx = -2 (impossible) Maths revision course by Miriam Hanks

  14. Solving trig equations • If there is only one of sin cos or tan, then solve by doing the inverse function: • sin x = ½ give answers 0 < x < x = sin-1(½) x = or √ √ (π – x)S A T C Maths revision course by Miriam Hanks

  15. Trigonometry –The wave function If you have a sin x and a cos x and a constant, it is not possible to solve the equation using the above techniques, so we use the wave function to reduce it to a single trig function eg sin x + 5cos x = 6 or 3sin x – 2cos x = -1 or ½cos x + 7sin x – 2= 0 Maths revision course by Miriam Hanks

  16. Trigonometry –The wave function eg Express 3 cos x – 4 sin x into the form k cos (x + a) Expand k cos (x + a) using the formula sheet first Maths revision course by Miriam Hanks

  17. Trigonometry –The wave function k cos (x + a) = kcos x cos a - k sin x sin a Then put this equal to the function in the question: 3 cos x – 4 sin x = kcos x cos a – k sin x sin a Maths revision course by Miriam Hanks

  18. Trigonometry –The wave function 3 cos x – 4 sin x = kcos x cos a – k sin x sin a Now compare how many cos x’s you have on each side of the equation: 3 = k cos a Now compare how many sin x’s you have on each side of the equation: 4 = k sin a Maths revision course by Miriam Hanks

  19. Trigonometry –The wave function 3 = k cos a 4 = k sin a Use to divide these equations: tan a = 4/3 a = tan-1 (4/3) a = 0.927 radians (to 3 dps) Maths revision course by Miriam Hanks

  20. Trigonometry –The wave function 3 = k cos a 4 = k sin a Use Pythagoras to find k: k = 32 +42 k = 5 So, the answer is 5 cos (x + 0.927 ) Maths revision course by Miriam Hanks

  21. Trigonometry –Max & min values If you are asked for a maximum or minimum value and the question is only worth one point, then use the fact that sin of anything and cos of anything have max & min values of 1 and –1. eg What are the maximum and minimum values of y when y = 2 + 3cos(x – 40). Answer: Since cos(x – 40) has max value 1, then the max value of y is 5. What is the min? Maths revision course by Miriam Hanks

  22. Trigonometry in real life When is trigonometry used in real life? Tidal experts are meteorologists use sine waves to predict tides. These are invaluable to sailors and kayakers when planning a trip. Maths revision course by Miriam Hanks

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