GR 10 MATHS TRIG RATIOS & FUNCTIONS

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GR MATHS TRG RATS & FUNCTNS Summar of TRG RATS Gr Maths Trig Ratios & Functions The SGNS of the trig ratios N A FLASH!

tan is PSTVE in quadrants & 3 (and negative in & 4) The 4 steps to find the trig ratios of an angle:. Place the ø in STANDARD PSTN (starting at X ).

Appl the DEFNTNS sin = cos = tan = r r sin 'SUNRSE FR SNE' cos 'PULL A CURTAN FR CS' tan V Apparentl, this is THE BMW SYMBL!

1 GR MATHS TRG RATS & FUNCTNS Summar of TRG RATS Gr Maths Trig Ratios & Functions The SGNS of the trig ratios N A FLASH! sin = r and is positive in & sin is PSTVE in quadrants & (and negative in 3 & 4) cos = r and is positive in & V cos is PSTVE in quadrants & 4 (and negative in & 3) tan = and & have the same sign in & tan is PSTVE in quadrants & 3 (and negative in & 4) The 4 steps to find the trig ratios of an angle:. Place the ø in STANDARD PSTN (starting at X ).... Pick a point (; ) on the end arm of the ø we'll call its distance from the origin r 3. Write down = = r = 4. Appl the DEFNTNS sin = cos = tan = r r sin 'SUNRSE FR SNE' cos 'PULL A CURTAN FR CS' tan V Apparentl, this is THE BMW SYMBL!!! end arm beginning arm (hpotenuse) r (adjacent) Learn these eas PCTURES so that ou know the SGNS of our trig ratios N A FLASH! N MRE CAST RULE!!! X (; ) (opposite) The trig ratios of 9º and multiples of 9º Use this procedure to find the trig ratios of 9º ; 8º ; 7º & 36º (& º) r = 5 (; 5) 9º sin 9º = r = 5 5 = cos 9º = r = 5 = tan 9º = = 5 = = ; = 5 ; r = 5 = -6 ; = ; r = 6 7º (; -4) r = 4 sin 7º = = -4 r 4 = cos 7º = = r 4 = tan 7º = = -4 (-6; ) 36º = ; = -4 ; r = 4 = 5 ; = ; r = 5 Note: The results for º and 36º are the same. r = 6 8º r = 5 (5; ) sin 8º = r = 6 = cos 8º = r = -6 6 = tan 8º = = -6 = sin 36º = r = 5 = 5 cos 36º = = r 5 = tan 36º = = 5 = : sin : cos : tan : ± ± SUMMARY The Answer Series Maths stud guides offer a ke to eam success. n particular, Gr Maths 3 in provides a superb foundation for the major topics in Senior Maths. Copright The Answer

TRG FUNCTNS Gr Maths Trig Ratios & Functions Trigonometric graphs We will learn how to sketch the graphs = sin, = cos and tan for º 36º. We will use the critical values of these ratios to make it eas.

2 TRG FUNCTNS Gr Maths Trig Ratios & Functions Trigonometric graphs We will learn how to sketch the graphs = sin, = cos and tan for º 36º. We will use the critical values of these ratios to make it eas. But first, some terminolog... Terminolog The sine and cosine graphs are WAVE-shaped. and now, these values on a number line: sin : cos : t is clear that sin and cos can onl be PRPER FRACTNS or equal to ± or. The amplitude of a WAVE is the deviation from its centre line: The period of a graph is the number of degrees spanning a FULL WAVE. The range is the set of all the possible -values. ur investigations of the trig ratios have shown us that the range of values of sines and cosines is ver small - onl between and. We write: sin and cos for all values of! B contrast, the range of tan values is from - to +! Before drawing the graphs, we will depict the 'critical values' of the ratios as the angle increases from º to 36º as: The graphs of = sin & = cos for [º; 36º] Use the wheels to plot the 'critical points' before drawing the waves. = sin = cos (9º; ) Note the - TURNNG PNTS - (º; ) - 9º 8º 7º 36º (7º; ) For both graphs: the amplitude = unit and the period = 36º. The range: sin The range: cos - 9º 8º 7º 36º (8º; ) (36º; ) ` 'Wheels' of values As : º 36º sin : cos : sin cos n these 'wheels' of values we are considering angles from º to 36º, going anticlockwise from the line X. We read the 'critical values'; i.e. the sine and cosine values of multiples of 9º accordingl, as indicated on the wheels. The Critical Values of = tan The range of tan values is (- ; )! So, we need tan values 'more often' than for sine and cosine. tan 35º = tan 5º = + tan 35º = Remember: 45º tan 45 = 45º... quadrant... quadrant... quadrant V 35 Check these values on our calculator. Also, confirm them b placing each angle in standard position V Use the definition tan = Copright The Answer

3 (- ; ) As : º 36º tan : The graph of = tan for [º; 36º] = tan = ; = â tan 35º = = 35º (; ) 5º = ; = â tan 5º = = tan + 45º 45º 9º 35º 8º 5º 7º 35º 36º = 9º = 7º Copright The Answer 3 35º (; ) = ; = â tan 35º = = The dashed lines, = 9º and = 7º, are called asmptotes The range: (- ; ) There is no amplitude, but the period of this graph is 8º. An asmptote is a line which a curve approaches but will never touch or cut. The Quadrants Gr Maths Trig Ratios & Functions We have observed the relationship between angles (º to 36º) and their trigonometric ratios (sin, cos, tan ), both on the 'wheels' and on the graphs. The wheels * We observe the angles, and * write down the ratio values. The graphs * We write down the values of the angles on the -ais, and * observe the values of the ratios. See where the quadrants lie in both cases. The quadrants: V The intervals: º 9º 9º 8º 8º 7º 7º 36º sin : cos : tan : Note where sin > (and where it is negative) Note where cos > (and where it is negative) º Note where tan > (and where it is negative) V V V V V V = sin = cos = tan

An investigation Use our calculator to observe various values of sin, cos and tan. Get to know how each of these three ratios behave as an angle increases, through each quadrant, from º to 36º.

Draw the graph of = sin on the following set of aes:,5 -,5 45º 9º 35º 8º 5º 7º 35º 36º Compare the values what do ou notice?

Draw the graph of = cos on the following set of aes: nvestigating the values of tan for [º; 36º] 3.

4 An investigation Use our calculator to observe various values of sin, cos and tan. Get to know how each of these three ratios behave as an angle increases, through each quadrant, from º to 36º. nvestigating the values of sin for [º; 36º]. Fill in the sin values (correct to decimal digits) of these angles º º 45º 7º 9º º 35º 6º 8º sin 8º º 5º 5º 7º 9º 35º 34º 36º sin. Draw the graph of = sin on the following set of aes:,5 -,5 45º 9º 35º 8º 5º 7º 35º 36º Compare the values what do ou notice? Use the accompaning table to plot points, even approimatel, on the set of aes. Gr Maths Trig: An investigation_questions. Draw the graph of = cos on the following set of aes: nvestigating the values of tan for [º; 36º] 3. Fill in the tan values (correct to decimal digits) of these angles º º 45º 7º 9º º 35º 6º 8º tan 8º º 5º 5º 7º 9º 35º 34º 36º tan,5 -,5 45º 9º 35º 8º 5º 7º 35º 36º 3. What is happening at 9º? And at 7º? We mabe need to investigate more values, namel those just before and after 9º and 7º: Again, compare the values what do ou notice? Round off to the nearest whole number 8º 85º 89º 89,9º 9º 9,º 9º 95º º tan ? º 65º 69º 69,9º 7º 7,º 7º 75º 8º tan ? nvestigating the values of cos for [º; 36º]. Fill in the cos values (correct to decimal digits) of these angles º º 45º 7º 9º º 35º 6º 8º cos 8º º 5º 5º 7º 9º 35º 34º 36º cos Compare the values what do ou notice? (Also compare the values of. vs. Use the accompaning table to plot points, even approimatel, on the set of aes. Copright The Answer Draw the graph of = tan on the following set of aes: 45º 9º 35º 8º 5º 7º 35º 36º - Use the accompaning table to plot points, even approimatel, on the set of aes.

4. ncrease / Decrease Now, use the tables and/or the graphs to fill in the spaces below and circle the correct word. : º 9º : 9º 8º : 8º 7º : 7º 36º Quadrant number 5.

Asmptotes The equations of the asmptotes of the graph = tan :.. Gr Maths Trig: An investigation_questions 9. Function notation f f() = sin ; g() = cos and h() = tan then f(º) =.......... ; g(º) =.

5 4. ncrease / Decrease Now, use the tables and/or the graphs to fill in the spaces below and circle the correct word. : º 9º : 9º 8º : 8º 7º : 7º 36º Quadrant number 5. Positive / Negative sin cos tan V V sin positive sin negative cos positive cos negative tan positive tan negative 6. Maimum / Minimum Maimum value Minimum value sin cos tan 8. Asmptotes The equations of the asmptotes of the graph = tan :.. Gr Maths Trig: An investigation_questions 9. Function notation f f() = sin ; g() = cos and h() = tan then f(º) = ; g(º) = and h(º) = f(9º) = ; g(8º) = and h(35º) = Solving Equations Solve the following equations where º 36º, correct to the nearest whole number. Remember: Use the tables and graphs in, and 3. Solve for : (a) sin = (b) sin = (c) sin = (d) sin =,34 (e) sin = -,34 (f) sin =,9 (g) sin =,94 (h) sin = -,94 (i) sin =,3. Solve for : (a) cos = (b) cos = (c) cos = (d) cos =,34 (e) cos = -,34 (f) cos =,9 (g) cos =,94 (h) cos = -,94 (i) cos =,3.3 Solve for : (a) tan = (b) tan = (c) tan = (d) tan =,36 (e) tan = -,36 (f) tan =,75 (g) tan = -,75 (h) tan = 57 (i) tan = -57 (j) tan is undefined when =? (h) & (i): correct to decimal digit 7. Features of the graphs = sin = cos = tan Amplitude Period Range Copright The Answer 5

6 EXERCSE 6.8 Eploring the role of a and q in trigonometric functions QUESTNS Gr Maths Trig Functions: Eercise_Questions Establish for each graph: the amplitude the range the period Figure (for A) Figure (for B) - - See figures to 6 alongside where the following graphs are drawn for : º 36º: Figure & both show the graph = sin ; Figure 3 & 4 both show the graph = cos and Figure 5 & 6 both show the graph = tan Sketch the following graphs on the given sets of aes: A. = sin C. = -cos E. = tan B. = sin + D. = cos - F. = tan + Plot each point - using the sin, cos & tan 'wheels' and not a calculator! Figure 3 (for C) Figure 4 (for D) sin : cos : tan : º Compare: A and B to the given graph of = sin in figures & ; C and D to the given graph of = cos in figures 3 & 4; and E and F to the given graph of = tan in figures 5 & 6 Copright The Answer 6 For A & B Amplitude Range Period For C & D Amplitude Range Period For E & F Amplitude Range Period - - = sin = sin = sin + = -cos = cos = cos - = tan = tan +

- - 36º - 9 8 7 36 9 8 7 36 = sin - - - 36º = cos - - 36º = cos + 3 3 36º = -cos 36º = tan cr 8º = 9º & = 7º - = -tan cr 8º = 9º & = 7º = tan - cr 8º = 9º & = 7º Stud the 'variations' carefull once

7 Gr Maths Trig: Eercise 6.8_Questions Figure 5 (for E) Figure 6 (for F) The following table will help ou make conjectures about the variations: amplitude range ma value min value period asmptotes = 3 sin º = - sin º = sin º = cos º = cos º = -cos 36º = tan cr 8º = 9º & = 7º - = -tan cr 8º = 9º & = 7º = tan - cr 8º = 9º & = 7º Stud the 'variations' carefull once ou've drawn the graphs ourself b plotting points. n each case, notice how the graph was different to the basic graph, i.e. = sin vs. = sin, etc. The aim of the eercise has been for ou to understand the effect of the values of a and q - the parameters - in the graphs: = asin + q = a cos + q = a tan + q This package is an etract from our Gr Maths 3 in stud guide. We trust that this will help ou to grow in confidence as ou prepare for our eams. The Answer Series stud guides have been the ke to eam success for man learners. Visit our website to find appropriate resources for our success! Copright The Answer 7

Trig Graphs. What is a Trig graph? This is the graph of a trigonometrical function e.g.

Trig Graphs. What is a Trig graph? This is the graph of a trigonometrical function e.g. Trig Graphs What is a Trig graph? This is the graph of a trigonometrical function e.g. sin, cos or tan How do we draw one? We make a table of value using the calculator. Tr to complete the one below (work