PROGRAMME F 8 TRIGONOMETRY STROUD Worked examples and

PROGRAMME F 8 TRIGONOMETRY STROUD Worked examples and exercises are in the text

Programme F 8: 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 F 8: Trigonometry Angles Rotation Radians Triangles Trigonometric ratios Reciprocal ratios Pythagoras’ theorem Special triangles STROUD Worked examples and exercises are in the text

Programme F 8: 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 (360 o ) One degree = 60 minutes (60'), STROUD and one minute = 60 seconds (60'') Worked examples and exercises are in the text

Angles: Radians Programme F 8: Trigonometry 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 Since circumference has total length 2 π r, there are 2π radians in a full circle. So 1 radian is 360/ 2π degrees, i. e. about 57 degrees. STROUD Worked examples and exercises are in the text

Useful Numbers of Radians and Degrees 2 π radians = 360 degrees π radians = 180 degrees π/2 radians = 90 degrees π/3 radians = 60 degrees π/4 radians = 45 degrees π/6 radians = 30 degrees 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 F 8: 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 STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Angles Trigonometric ratios STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Trigonometric ratios: in a right-angled triangle AB = the “hypotenuse” ERROR in tangent formula: should be AC/BC !!!! STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Reciprocal ratios STROUD Worked examples and exercises are in the text

Programme F 8: 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 STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Special triangles Right-angled isosceles STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Special triangles, contd Half equilateral STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Angles Special triangles Half equilateral STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Angles Trigonometric identities Trigonometric formulas STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Trigonometric identities The fundamental identity Two more identities Identities for compound angles STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry The fundamental identity The fundamental trigonometric identity is derived from Pythagoras’ theorem STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Two more identities Dividing the fundamental identity by cos 2 STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Last one: Dividing the fundamental identity by sin 2 STROUD Worked examples and exercises are in the text

Beyond Pythagoras Cosine Rule (added by John Barnden) Ignore the outer triangle. Let the sides of the inner triangle ABC have lengths a, b, c (opposite the angles A, B, C, respectively). Then: c 2 = a 2 + b 2 – 2 ab. cos C This works for any shape of triangle. When C = 90 degrees, we just get Pythagoras, as cos 90 o = 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) Sine Rule 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 F 10: Functions Switching to Programme F 10 briefly STROUD Worked examples and exercises are in the text

Programme F 10: 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 F 10: Functions Trigonometric functions: Rotation beyond 90 degrees os can i t a r c i ometr n o g i r t ke AB he t a e T l. c r e i l y ang unit c n a the a f o r t o o f f s i u d i e i l d v ositi s, va he ra t p n o e i B t t a c O t n hown o e s u r k f e o a c t s i T a r g t. c n tinui nome ve in below i n o t f o g i a i c r g e t e y v i e n t B th. , OB g into th e line, nega e n v d i e t e l i d s e n o exte ositiv ve th AB p p o o s b S a a. h f t i f s e lway the le a o ) t positiv s e u i v i d egat e ra right, n ypotenuse (th H below. STROUD Worked examples and exercises are in the text

Programme F 10: Functions Trigonometric functions Rotation The sine function: STROUD Worked examples and exercises are in the text

Programme F 10: Functions Trigonometric functions Rotation The cosine function: STROUD Worked examples and exercises are in the text

Programme F 10: Functions Trigonometric functions The tangent is the ratio of the sine to the cosine: STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Switching back to Programme F 8 STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Angles Trigonometric identities Trigonometric formulas STROUD Worked examples and exercises are in the text

Programme F 8: 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

Trigonometric formulas Programme F 8: Trigonometry Sums and differences of angles (NB: there’s a typo on LHS of 2 nd sine formula – should have a minus sign instead of a plus sign – John B. ) STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Double angles STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Sums and differences of trig functions STROUD Worked examples and exercises are in the text

Programme F 8: Trigonometry Products of trig functions STROUD Worked examples and exercises are in the text