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1 Higher Mathematics CfE Edition This document w as produced specially for the website, and we require that any copies or derivative works attribute the work to Higher Still Notes. For more details about the copyright on these notes, please see hsn . Trigonometry Contents Trigonometry 1 1 Radians EF 1 2 Exact Values EF 1 3 Solving Trigonometric Equations RC 2 4 Trigonometry in Three Dimensions EF 5 5 Compound Angles EF 8 6 Double-Angle Formulae EF 11 7 Further Trigonometric Equations RC 12 8 Expressing pcosx + qsinx in the form kcos(x a) EF 14 9 Expressing pcosx + qsinx in other forms EF 15 10 Multiple Angles EF 16 11 Maximum and Minimum Values EF 17 12 Solving Equations RC 18 13 Sketching Graphs of y = pcosx + qsinx EF 20 Higher Mathematics Trigonometry Page 1 CfE Edition hsn.

2 Trigonometry 1 Radians EF Degrees are not the only units used to measure angles. The radian (RAD on the calculator) is a measurement also used. Degrees and radians bear the relationship: radians 180 .= The other equivalences that you should become familiar with are: 630 radians = 445 radians = 360 radians = 290 radians = 34135 radians = 3602 radians. = Converting between degrees and radians is straightforward. To convert from degrees to radians, multiply by and divide by 180. To convert from radians to degrees, multiply by 180 and divide by . For example, 5180185050 radians = =. 2 Exact Values EF The following exact values must be known.

3 You can do this by either memorising the two triangles involved, or memorising the table. DEG RAD sin x cos x tan x 0 0 0 1 0 30 6 12 32 13 45 4 12 12 1 60 3 32 12 3 90 2 1 0 1 1 1 2 Tip You ll probably find it easier to remember the triangles. Degrees Radians 180 180 Higher Mathematics Trigonometry Page 2 CfE Edition hsn . 3 Solving Trigonometric Equations RC You should already be familiar with solving some trigonometric equations. EXAMPLES 1. Solve 12sinx = for 0360x<<. 12sinx = 180180360 SATC xxxx + Since sinx is positive First quadrant solution: ( ) == 30 or 180 3030 or = 2. Solve 15cosx = for 0360x<<.

4 15cosx = 180360180 SATC xxxx + Since cosx is negative ()115cos63 435 (to 3 ).x == 180 63 435 or 180 63 435116 565 or 243 += 3. Solve sin3x = for 0360x<<. There are no solutions since 1 sin1x . Note that 1 cos1x , so cos3x = also has no solutions. Remember The exact value triangle: 1 2 330 60 Higher Mathematics Trigonometry Page 3 CfE Edition hsn . 4. Solve tan5x = for 0360x<<. tan5x = 180180360 SATC xxxx + Since tanx is negative ( )1tan578 690 (to 3 ).x == 180 78 690 or 360 78 690101 310 or 281 = Note All trigonometric equations we will meet can be reduced to problems like those above.

5 The only differences are: the solutions could be required in radians in this case, the question will not have a degree symbol, Solve 3 tan1x= rather than 3 tan1x = ; exact value solutions could be required in the non-calculator paper you will be expected to know the exact values for 0, 30, 45, 60 and 90 degrees. Questions can be worked through in degrees or radians, but make sure the final answer is given in the units asked for in the question. EXAMPLES 5. Solve 2 sin 21 0x = where 0360x . 122 sin 21sin 2xx = = 1802218023602 SATC xxxx + 03600 2720xx ( ) == 230 or 180 30or 360 30 or 360 180 30or 360 360 30x= ++ ++230 or 150 or 390 or 51015 or 75 or 195 or Note There are more solutions every 360 , since sin(30 ) = sin(30 +360 ) =.

6 So keep adding 360 until 2x > 720. Remember The exact value triangle: 1 2 330 60 Higher Mathematics Trigonometry Page 4 CfE Edition hsn . 6. Solve 2 cos 21x= where 0x . 12cos 2x= 22222 SATC xxxx + 002 2xx ( ) == 4442 or 2or 2x = +7447882 or or .xx == 7. Solve 24 cos3x= where 02x <<. ()2343432coscoscosxxx== = SATC Since cosx can be positive or negative ( ) == 666 66 or or or 2or 2x = + +571166 6 6 or or or .x = 8. Solve ()3 tan 3205x = where 0360x . ()()533 tan 3205tan 320xx = = SATC 03600 3108020 320 1060xxx ( )153320 tan59 036 (to 3 )x == 320 59 036 or 180 59 036or 360 59 036 or 360 180 59 036or 360 360 59 036 or 360 360 180 59 036or 360 360 360 59 =++++++++++++ Remember The exact value triangle: 1 1 4 24 Remember The exact value triangle: 1 2 36 3 Higher Mathematics Trigonometry Page 5 CfE Edition hsn.

7 320 59 036 or 239 036 or 419 036or 599 036 or 779 036 or 959 036379 036 or 259 036 or 439 036or 619 036 or 799 036 or 979 03626 35 or 86 35 or 146 35 or 206 35 or 266 3xxx ===5 or 326 35. 9. Solve ()3cos 20 812x += for 02x <<. ()3cos 20 812x += SATC <<<<< +< +< +<33330202 4241 047 213 614 (to 3 )xxxx ()132cos0 8120 623 (to 3 ).x +== 320 623x += or 20 623or 20 623 or 220 623or 220 623 or 2220 623 ++ ++++ ..25 660 or 6 906 or 11 943 or ..24 613 or 5 859 or 10 896 or 307 or 2 930 or 5 448 or 6 +=== 4 Trigonometry in Three Dimensions EF It is possible to solve trigonometric problems in three dimensions using techniques we already know from two dimensions.

8 The use of sketches is often helpful. The angle between a line and a plane The angle a between the plane P and the line ST is calculated by adding a line perpendicular to the plane and then using basic trigonometry. PSTaRemember Make sure your calculator uses radians Higher Mathematics Trigonometry Page 6 CfE Edition hsn . EXAMPLE 1. The triangular prism ABCDEF is shown below. Calculate the acute angle between: (a) The line AF and the plane ABCD. (b) AE and ABCD. (a) Start with a sketch: ( ) 699 (or 0 291 radians) (to 3 d p ).aa ==== .. Note Since the angle is in a right-angled triangle, it must be acute so there is no need for a CAST diagram.

9 (b) Again, make a sketch: We need to calculate the length of AC first using Pythagoras s Theorem: 22AC106136=+= () 426 (or 0 252 radians) (to 3 d p ).bb ==== .. 136 cmTherefore: bAEC3 cmCDA10 cm6 cmangle to be calculated bAEC3 cmAa3 cm10 cmFDBAFECD6 cm3 cm10 cmHigher Mathematics Trigonometry Page 7 CfE Edition hsn . The angle between two planes The angle a between planes P and Q is calculated by adding a line perpendicular to Q and then using basic trigonometry. EXAMPLE 2. ABCDEFGH is a cuboid with dimensions 12 8 8 cm as shown below. (a) Calculate the size of the angle between the planes AFGD and ABCD.

10 (b) Calculate the size of the acute angle between the diagonal planes AFGD and BCHE. (c) Start with a sketch: ( ) 690 (or 0 588 radians) (to 3 d p ).aa ==== .. (d) Again, make a sketch: Let AF and BE intersect at T ATB is isosceles, so == .T AB TBA 33 690 ()..A TB 18033 69033 620 .= + = So the acute angle is: .BTFA TE 180112 380 (or 1 176 radians) (to 3 d p ).== = .. A B F E T ABF12 cm8 cmaABCDEFGH8 cm8 cm12 cmaPQNote The angle could also have been calculated using rectangle DCGH. Note Angle GDC is the same size as angle FAB. Higher Mathematics Trigonometry Page 8 CfE Edition hsn . 5 Compound Angles EF When we add or subtract angles, the result is a compound angle.