# Trigonometric Functions and Graphs

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Trigonometric Functions and Graphs
Higher Mathematics Unit 1 Trigonometric Functions and Graphs

RADIANS So far we have always measured our angles in degrees.
There is another way to measure angles. It is particularly important in applied mathematics. Angles can be measured in RADIANS

AOB subtends an arc equal to a radius

2 2 Every 90° is radians

2 90° 180° 2 3 2 270°

3 60° = radians (as 180°  ⅓ = 60°)

We can also convert as follows. 180 Degrees Radians Convert 150° to radians (simplifying fraction: divide by 30) 150   180 5  6 Radians

60   180 2 3 Radians 120   180 7 6 Radians 210   180 7 4 Radians 315   180

4  radians = 180° 4 4 180° radians = = 45° 3 2 Change radians to degrees 3 2 radians = = 270° 2 3 180°

We can also convert as follows. 180 Radians Degrees 5 6 Radians Convert to degrees 5   180 6   5  180 6 150°

Change to degrees   180  45° Radians 2 Radians 120° 3 135°
4   45° 2 3 Radians 2   180 3   120° 3 4 Radians 3   180 4   135° 5   180 3   5 3 Radians 300°

 radians = 180°  The angles in the following table must be known.
They are essential for non-calculator questions. Degrees 360 180 90 60 45 30 2 2 3 4 6 Radians remember as factors or multiples of 180°  radians = 180°

Most angles in non-calculator work are multiples of those above
Use them to complete the table below Degrees 120° Use for Christines Use smartboard document radians to angles doc 135° 210° 270° 315° 360° 2 3 5 6 5 4 4 3 5 3 Radians

Degrees Radians 120° 135° 150° 210° 225° 240° 270° 300° 315° 360° 2 3
Use for Christines only 135° 150° 210° 225° 240° 270° 300° 315° 360° 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4 11 6 Radians

Sketching Trig Graphs For reinforcement and using the graphs go to folder graph transformations

Trig Graphs The maximum value for sin x is 1 when x = 90°
The minimum value for sin x is when x = 270° sin x = 0 (i.e. cuts the x-axis) at: x = 0°, x = 180°, x = 360

Trig Graphs  The maximum value for sin x is 1 when x =
The minimum value for sin x is when x = 2 3 2 sin x = 0 (i.e. cuts the x-axis) at: x = x = x = 2

y = asinx

Trig Graphs When sin x is multiplied by a number, that number gives the maximum and minimum value of the function. Note the function still cuts the x-axis at: x = 0,  & 2

Trig Graphs The maximum value for cos x is 1 when x = 0° & 360°
The minimum value for cos x is when x = 180° cos x = 0 (i.e. cuts the x-axis) at: x = 90°, x = 270°

Trig Graphs The maximum value for cos x is 1 when x = 0 & 2
The minimum value for cos x is when x =  cos x = 0 (i.e. cuts the x-axis) at: x = x = 2 3 2

y = acosx

Trig Graphs When cos x is multiplied by a number, that number gives the maximum and minimum value of the function. Note the function still cuts the x-axis at: x = 2 3 2

Trig Graphs Using radians, sketch the following trig graphs: y = 5sinx
y = 1.5cosx y = 2cosx y = 100sinx Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

y = -sinx

y = -sinx The function y = -sinx is a reflection of y = sinx in the x - axis.

y = -cosx

y = -cosx The function y = -cosx is a reflection of y = cosx in the x - axis.

y = sin nx

y = sin nx

Trig Graphs PERIOD PERIOD When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p. i.e. for y = sin nx: period of graph = 2p n

y = cos nx

Trig Graphs PERIOD PERIOD When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p. i.e. for y = cos nx: period of graph For support of classwork use exercises on smartboard document trig transformations = 2p n

Trig Graphs Using radians, sketch the following trig graphs:
y = 5sin2x y = 4cos2x y = 6cos3x y = 7sin½x Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

Adding or subtracting from a Trig Function

Trig Graphs When a number is added to a trig function the graph “slides” vertically up by that number. When a number is subtracted from a trig function the graph “slides” vertically down by that number. For support of classwork use exercises on smartboard document trig transformations

Trig Graphs Using radians, sketch the following trig graphs:
y = 3 + sin2x y = cos3x - 4 y = 3sinx + 2 y = 2cos2x - 1 y = 2 - sinx Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

Trig Graphs When a number is added to x the graph “slides” to the left by that number. When a number is subtracted from x the graph “slides” to the right by that number by that number. For support of classwork use exercises on smartboard document trig transformations

Example 1 Find the maximum turning point, for 0 ≤ x ≤ p, of the graph y = 5sin(x - p/3). Consider the function y = 5sin x Maximum value is 5 When x = p/2 For y = 5sin(x - p/3) Max occurs at p/2 + p/3 = 5p/6 Turning Point: (5p/6,5)

Example 2 Write down the equation of the drawn function and the period of the graph. Write the function as y = asin bx + c

Example 2 y = asin bx + c b = 3 (3 wavelengths in 2p) Period of graph 2p  3 = 2p/3 Difference between max and min = 12 a = 12  2 = 6 y = 6sin 3x + c (graph then shifts up 2) c = +2 y = 6sin 3x + 2

Ratios and Exact Values
Exact Values for 45° 1 Square 1 45° 1 45°

Ratios and Exact Values
Exact Values for 45° 1 45° x² = 1² + 1² x² = 2 x = √2 x √2

Ratios and Exact Values
Exact Values for 45° 1 √2 Sin 45° = Cos 45° = Tan 45° = 1 45° √2 1 √2 1

Ratios and Exact Values
Exact Values for p/4 1 √2 Sin p/4 = Cos p/4 = Tan p/4 = 1 p/4 √2 1 √2 1

Ratios and Exact Values
Exact Values for 30° & 60° 60° 30° 2 1 60° 30° 1 2 60° 2 Equilateral Triangle

Ratios and Exact Values
Exact Values for 30° & 60° 60° 30° 1 2 x² = 2² - 1² x² = 3 x = √3 x √3

Ratios and Exact Values
Exact Values for 30° 60° 30° 1 2 1 2 Sin 30° = Cos 30° = Tan 30° = √3 2 √3 1 √3

Ratios and Exact Values
Exact Values for p/6 p/3 p/6 1 2 1 2 Sin p/6 = Cos p/6 = Tan p/6 = √3 2 √3 1 √3

Ratios and Exact Values
Exact Values for 60° 60° 30° 1 2 Sin 60° = Cos 60° = Tan 60° = √3 2 1 2 √3 √3

Ratios and Exact Values
Exact Values for p/3 p/3 p/6 1 2 Sin p/3 = Cos p/3 = Tan p/3 = √3 2 1 2 √3 √3

Angles Greater than 90° SIN positive ALL Positive 2nd Quadrant 1st
Sin A = (+)ve Cos A = (-)ve Tan A = (-)ve 1st Quadrant Sin A = (+)ve Cos A = (+)ve Tan A = (+)ve 180° 3rd Quadrant TAN positive Sin A = (-)ve Cos A = (-)ve Tan A = (+)ve 4th Quadrant Sin A = (-)ve Cos A = (+)ve Tan A = (-)ve COS Positive 270°

Angles Greater than p/2 SIN positive ALL Positive TAN positive COS
TAN positive COS Positive 3p/2

sin 3p/4 positive cos 7p/6 negative tan 7p/4 cos 5.4 radians TAN
ALL Positive COS SIN 2p sin 3p/4 cos 7p/6 tan 7p/4 cos 5.4 radians positive negative For class support use smartboard document Four quadrants

90° SIN ALL (180 - x)° 180° 360° (180 + x)° (360 - x)° COS TAN 270°

Example 3 Solve 2sin x° = 1, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = sin x 2 sin x° = 1 sin x° = ½ For printing only see smatrboard document equations

Since sin x° is positive it is in the 1st and 2nd quadrants
Example 3 sin x° = ½ Since sin x° is positive it is in the 1st and 2nd quadrants For printing only see smatrboard document equations

Example 3 sin x° = ½ √3 sin x° = ½ sin 30° = ½ x = 30° 30° 2 60° 1
For printing only see smatrboard document equations 60° 1

Example 3 sin x° = ½ sin 30° = ½ x = 30° x = 30° or x = 180° - 30°
For printing only see smatrboard document equations

Example 3 For printing only see smatrboard document equations

p/2 SIN ALL (p - )  p 2p (p + ) (2p - ) COS TAN 3p/2

Example 4 Solve √2cos  +1 = 0, 0 ≤  ≤ 2 and illustrate the solution in a sketch of y = cos  √2cos  +1 = 0 √2cos  = -1 Cos  = √2 -1 For printing only see smatrboard document equations

Since cos  is negative it is in the 2nd and 3rd quadrants
Example 4 Cos  = Since cos  is negative it is in the 2nd and 3rd quadrants √2 -1 For printing only see smatrboard document equations

Example 4 Cos  = √2 cos  = cos /4 =  = /4 1 p/4 1 -1 √2 1 √2 1 √2
For printing only see smatrboard document equations 1 √2 1

Example 4 Cos  = cos  = cos /4 = = /4 So  =  - /4 or  + /4
√2 -1 √2 1 cos  = cos /4 = = /4 So  =  - /4 or  + /4 = 3/4 or 5/4 √2 1 For printing only see smatrboard document equations

Example 4 For printing only see smatrboard document equations

Consider if the equation was cos x = ½
Example 5 Solve cos 3x° = ½, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = cos 3x Consider if the equation was cos x = ½ As cos x is positive it must be in the 1st and 4th quadrants. For printing only see smatrboard document equations

Example 5 √3 cos x = ½ Cos 60° = ½ x = 60° or 360° - 60°
30° 2 √3 cos x = ½ Cos 60° = ½ x = 60° or 360° - 60° x = 60° or 300° For printing only see smatrboard document equations 60° 1

However the function we are using is cos 3x
Example 5 However the function we are using is cos 3x Therefore if x = 60° or 300° for cos x = ½ 3x = 60° or 3x = 300°: x = 20° or 100° the graph repeats itself 3 times in 360° with a wavelength of 120° as the function has a wavelength of 120° x = 20° or 100° or 140° or 220° or 260° or 340° For printing only see smatrboard document equations

Example 5 For printing only see smatrboard document equations

and negative, x will be in all four quadrants
Example 6 Solve 2sin² x° = 1 sin² x = ½ sin x = √½ sin x =  As sin x is positive and negative, x will be in all four quadrants √2 1 For printing only see smatrboard document equations

Example 6 sin x = sin 45° = x = 45° √2 1 45° 1 1 √2 1 √2
For printing only see smatrboard document equations 45° 1

Example 6 sin x = x = 45° or 180° - 45° or 180° + 45° or 360° - 45°
√2 1 For printing only see smatrboard document equations

Factorise the equation Consider the equation as: 4x² + 11x + 6 =
Example 7 Solve 4sin²  + 11sin  + 6 = 0, correct to 2 decimal places, for 0 ≤  ≤ 2 Factorise the equation Consider the equation as: 4x² + 11x + 6 = (4x + 3)(x + 2) = 0 4sin²  + 11sin  + 6 = 0 (4sin + 3)(sin + 2) = 0 For printing only see smatrboard document equations

sin = or sin = -2 (no solution)
Example 7 4sin²  + 11sin  + 6 = 0 (4sin + 3)(sin + 2) = 0 4sin + 3 = 0 or sin+ 2 = 0 4sin = -3 or sin = -2 sin = or sin = -2 (no solution) Therefore we have to solve sin = -0.75 4 -3 For printing only see smatrboard document equations

As sin is negative answer must be in 3rd and 4th quadrants
Example 7 sin = -0.75 As sin is negative answer must be in 3rd and 4th quadrants sin = 0.75  = sin-¹0.75 (radians)  = 0.85 radians For printing only see smatrboard document equations

Example 7  = 0.85 radians =  + 0.85 or  = 2 - 0.85
= 3.99 or 5.43 radians For printing only see smatrboard document equations

Reminders: sin² x° + cos² x° = 1 sin² x° = 1 - cos² x° cos² x° = 1 - sin² x°

Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360
Example 8 Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360 (substitute cos² x° = 1 - sin²x° into the equation) 1 - sin² x° + sin x° = 1 1 - sin² x° + sin x° -1 = 0 sin x° - sin² x° = 0 sin x°(1 - sin x°) = 0 sin x° = 0 or 1 - sin x° = 0 sin x° = 1 x = 0°or 180° or 360° or x = 90° For printing only see smatrboard document equations

Consider if the equation was sin x = 0.6 x = 36.87 or 180 - 36.87
Example 9 Solve sin (2x - 20)° = 0.6, correct to 1 decimal place, for 0 ≤ x ≤ 360 Consider if the equation was sin x = 0.6 x = or x = 36.87° or ° For printing only see smatrboard document equations

The function we are considering is sin (2x - 20)
Example 9 x = 36.87° or ° The function we are considering is sin (2x - 20) Therefore 2x - 20 = or 2x - 20 = , 2x = or 2x = x = 28.4° or x = 81.6° For printing only see smatrboard document equations

The function repeats itself twice in 360°
Example 9 The function repeats itself twice in 360° i.e. it has a wavelength of 180° x = 28.4° or x = 81.6° or x = ° or x = ° x = 28.4° or 81.6° or 208.4° or 261.6° For printing only see smatrboard document equations

Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤  ≤ 
Example 10 Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤  ≤  Consider if the equation was 3cos x = 1 cos x = ⅓  = 1.23 or 2 (remember to put calculator in radians)  = 1.23 or  = 1.23 or 5.05 radians For printing only see smatrboard document equations

The function we are considering is cos(2 + /4)
Example 10  = 1.23 or 5.05 radians The function we are considering is cos(2 + /4) 2 + /4 = or 2 + /4 = 5.05 2 = or 2 = 2 = or 2 = 4.26  = or  = 2.1 (to 1dp) Do not need to add on a wave length of  as ≤  ≤  For printing only see smatrboard document equations