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PROGRAMME F8 TRIGONOMETRY STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas NB: I have slightly edited the book’s slides STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Rotation Radians Triangles Trigonometric ratios Reciprocal ratios Pythagoras’ theorem Special triangles STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Rotation When a straight line is rotated about a point it sweeps out an angle that can be measured in degrees or radians A straight line rotating through a full angle and returning to its starting point is said to have rotated through 360 degrees (360o ) One degree = 60 minutes (60'), and one minute = 60 seconds (60'') STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Radians When a straight line of length r is rotated about one end so that the other end describes an arc of length r the line is said to have rotated through 1 radian – 1 rad STROUD Worked examples and exercises are in the text All at Sea (added by John Barnden) The circumference of the Earth is about 24,900 miles. That corresponds to 360 x 60 minutes of arc, = 21,600' So 1' takes you about 24,900/21,600 miles = about 1.15 miles. A nautical mile was originally defined as being the distance that one minute of arc takes you on any meridian (= line of longitude). This distance varies a bit as you go along the meridian, because of the irregular shape of the Earth. A nautical mile is now defined as 1852 metres, which is about 1.15 miles. A knot is one nautical mile per hour. NB: 60 knots is nearly 70 miles/hour. Look up nautical miles and knots on the web – it’s interesting. STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Triangles All triangles possess shape and size. The shape of a triangle is governed by the three angles and the size by the lengths of the three sides AB AC BC AB AC BC STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Trigonometric ratios AB AC BC AB AC BC so that: AB AB AB AB AC AC and and AC AC BC BC BC BC STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Trigonometric ratios: in a right-angled triangle AB = the “hypotenuse” sine of angle AC - denoted by sin AB cosine of angle BC - denoted by cos AB tangent of angle BC - denoted by tan AB Error in tangent formula: should be AC/BC STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Reciprocal ratios cosecant of angle secant of angle 1 - denoted by cos ec sin 1 - denoted by sec cos cotangent of angle STROUD 1 - denoted by cot tan Worked examples and exercises are in the text Programme F8: Trigonometry Pythagoras’ s Theorem The square on the hypotenuse of a rightangled triangle is equal to the sum of the squares on the other two sides a 2 b2 c2 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Special triangles Right-angled isosceles Angles measured in degrees: sin 45 cos 45 1 and tan 45 1 2 Angles measured in radians: 1 sin / 4 cos / 4 and tan / 4 1 2 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Special triangles, contd Half equilateral Angles measured in degrees: sin 30 cos 60 sin 60 cos30 tan 60 STROUD 1 2 3 2 1 3 tan 30 Worked examples and exercises are in the text Programme F8: Trigonometry Angles Special triangles Half equilateral Angles measured in radians: sin / 6 cos / 3 sin / 3 cos / 6 tan / 3 STROUD 1 tan / 6 1 2 3 2 3 Worked examples and exercises are in the text Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Trigonometric identities The fundamental identity Two more identities Identities for compound angles STROUD Worked examples and exercises are in the text Programme F8: Trigonometry The fundamental identity The fundamental trigonometric identity is derived from Pythagoras’ theorem a b c 2 2 2 so a 2 b2 2 1 2 c c that is: cos 2 sin 2 1 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Two more identities Dividing the fundamental identity by cos2 cos sin 1 2 2 so that cos 2 sin 2 1 cos 2 cos 2 cos 2 that is: 1 tan 2 sec 2 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Last one: Dividing the fundamental identity by sin2 cos sin 1 2 2 so that cos 2 sin 2 1 sin 2 sin 2 sin 2 that is: cot 2 1 cosec 2 STROUD Worked examples and exercises are in the text Beyond Pythagoras Ignore the outer triangle. (added by John Barnden) Let the sides of the inner triangle ABC have lengths a, b, c (opposite the angles A, B, C, respectively). Then: c2 = a2 + b2 – 2ab.cos C This works for any shape of triangle. When C = 90 degrees, we just get Pythagoras, as cos 90o = 0. EX: What happens when C is zero? STROUD Worked examples and exercises are in the text Beyond Pythagoras, contd The result on the previous slide can easily be shown be dropping a perpendicular from vertex A to line BC. Try it as an EXERCISE. Use Pythagoras on each of the resulting right-angle triangles. You’ll also need to use the Fundamental Identity. STROUD Worked examples and exercises are in the text Another Interesting Fact (added by John Barnden) a/sin A = b/sin B = c/sin C This again can easily be seen by dropping a perpendicular from any vertex to the opposite side. Try it as an EXERCISE. Just use the definition of sine twice to get two different expressions for the length of the perpendicular. STROUD Worked examples and exercises are in the text Programme F10: Functions Switiching to Programme F10 briefly – reminder of part of Bohnet’s coverage in Term 1 STROUD Worked examples and exercises are in the text Programme F10: Functions Trigonometric functions Rotation For angles greater than zero and less than /2 radians the trigonometric ratios are well defined and can be related to the rotation of the radius of a unit circle: STROUD Worked examples and exercises are in the text Programme F10: Functions Trigonometric functions Rotation By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle: STROUD Worked examples and exercises are in the text Programme F10: Functions Trigonometric functions Rotation The sine function: STROUD Worked examples and exercises are in the text Programme F10: Functions Trigonometric functions Rotation The cosine function: STROUD Worked examples and exercises are in the text Programme F10: Functions Trigonometric functions The tangent The tangent is the ratio of the sine to the cosine: STROUD tan sin cos Worked examples and exercises are in the text Programme F8: Trigonometry Switching back to Programme F8 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Angles Trigonometric identities Trigonometric formulas STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Trigonometric formulas Sums and differences of angles Double angles Sums and differences of ratios Products of ratios STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Trigonometric formulas Sums and differences of angles (NB: there’s a typo on LHS of 2nd sine formula – John B.) cos( ) cos cos sin sin cos( ) cos cos sin sin sin( ) sin cos cos sin sin( ) sin cos cos sin tan tan 1 tan tan tan tan tan( ) 1 tan tan tan( ) STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Double angles cos 2 cos 2 sin 2 cos 2 1 2sin 2 cos 2 2cos 2 1 sin 2 2sin cos tan 2 STROUD 2 tan 1 tan 2 Worked examples and exercises are in the text Programme F8: Trigonometry Sums and differences of trig functions sin sin 2sin cos 2 2 sin sin 2cos sin 2 2 cos cos 2cos cos 2 2 cos cos 2sin sin 2 2 STROUD Worked examples and exercises are in the text Programme F8: Trigonometry Products of trig functions 2sin cos sin( ) sin( ) 2cos cos cos( ) cos( ) 2sin sin cos( ) cos( ) STROUD Worked examples and exercises are in the text