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**Trigonometric Functions and Graphs**

Higher Mathematics Unit 1 Trigonometric Functions and Graphs

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**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

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**Radian Measure Length AB = radius**

AOB subtends an arc equal to a radius

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1 2 radians = 360° radians = 180°

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** radians = 180° So 1 radian = So 1 radian ~ 57° ~ 180° = radians**

2 2 Every 90° is radians

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2 90° 180° 2 0° 3 2 270°

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**180° = radians Degrees to radians Change 60° to radians:**

3 60° = radians (as 180° ⅓ = 60°)

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** 180 Radians Degrees Convert 150° to radians**

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

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**Change to Radians: 60° = 120° = Radians 210° = 2 315° = Radians**

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

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** radians = 180° Radians to Degrees: Change radians to degrees**

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

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** 180 Degrees Radians Convert to degrees Radians**

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

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**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°

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** 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°

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**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

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**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

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

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**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

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**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

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y = asinx

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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

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**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°

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**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

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y = acosx

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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

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**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

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y = -sinx

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

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y = -cosx

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

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y = sin nx

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y = sin nx

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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

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y = cos nx

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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

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**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

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**Adding or subtracting from a Trig Function**

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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

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**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

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**Adding or subtracting from x**

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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

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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)

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

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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

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**Ratios and Exact Values**

Exact Values for 45° 1 Square 1 45° 1 45°

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**Ratios and Exact Values**

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

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**Ratios and Exact Values**

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

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**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

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**Ratios and Exact Values**

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

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**Ratios and Exact Values**

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

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**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

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**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

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**Ratios and Exact Values**

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

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**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

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**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° 0° 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°

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

TAN positive COS Positive 3p/2

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**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

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

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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

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**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

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**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

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**Example 3 sin x° = ½ sin 30° = ½ x = 30° x = 30° or x = 180° - 30°**

For printing only see smatrboard document equations

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Example 3 For printing only see smatrboard document equations

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

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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

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**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

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**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

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**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

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Example 4 For printing only see smatrboard document equations

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**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

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**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

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**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

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Example 5 For printing only see smatrboard document equations

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**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

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**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

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**Example 6 sin x = x = 45° or 180° - 45° or 180° + 45° or 360° - 45°**

√2 1 For printing only see smatrboard document equations

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**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

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**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

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**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

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**Example 7 = 0.85 radians = + 0.85 or = 2 - 0.85**

= 3.99 or 5.43 radians For printing only see smatrboard document equations

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

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**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

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**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

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**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

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**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

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**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

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**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

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