MHF 4U - Unit 7 Trigonometric Identities and Equations - PDF

# MHF 4U - Unit 7 ## Trigonometric Identities and Equations ### 7.1 Exploring Equivalent Trigonometric Functions **Consider the following graph:** * **What are some possible equations for this graph?** * *f(x) = -sinx* - Reflected on the y-axis * *f(x) = cos(x + π/2)* - Left π/2 * *f...

# MHF 4U - Unit 7 ## Trigonometric Identities and Equations ### 7.1 Exploring Equivalent Trigonometric Functions **Consider the following graph:** * **What are some possible equations for this graph?** * *f(x) = -sinx* - Reflected on the y-axis * *f(x) = cos(x + π/2)* - Left π/2 * *f(x) = sin(x + π)* - Left π * *f(x) = -cos(x - π/2)* - Reflected on the x-axis, right π/2 **Since trigonometric functions are periodic, there are many equivalent trigonometric expressions.** **Learning Goal:** To be able to write an equivalent trigonometric function by using: 1. Horizontal Translations 2. Symmetry 3. Complementary Angles 4. Related Acute Angles **1. Horizontal Translations** * **Consider the graphs of sine and cosine below. Could you shift the graph left or right to produce the same graph?** * **Example:** Shifting *cos(x)* right π/2 produces the same graph. *Cos(x - π/2) = sin(x)* * **Example:** Shifting *sin(x)* right 2π produces the same graph. *Sin(x - 2π) = sin(x)* * **Example:** Shifting *cos(x)* right 2π produces the same graph. *Cos(x - 2π) = cos(x)* * **i) Horizontal translations of sine or cosine that are multiples of the period result in an equivalent expression.** * *sin θ = sin(θ + 2π)* * *cos θ = cos(θ + 2π)* * **ii) Horizontal translations of π/2 involving both the sine and cosine function result in an equivalent expression.** * *sin θ = cos(θ - π/2)* * *cos θ = sin(θ + π/2)* **2. Symmetry** * **Recall from earlier units that:** * **EVEN** functions have symmetry about the y-axis, so *f(x) = f(-x)* * **ODD** functions have rotational symmetry about the origin so *f(-x) = -f(x)* * **Determine when the following functions have even or odd symmetry. Use a sketch or graphing software to determine this.** | Function | Type of Symmetry | |----------------|--------------------| | *y = sin θ* | Odd Symmetry | | *y = cos θ* | Even Symmetry | | *y = tan θ* | Odd Symmetry | **Example 1** Write an expression that is equivalent to the following expressions. * **a) cos(5π/3 + 2π)** * = *cos(5π/3)* - **OR** - = *cos(-5π/3)* * = *cos(π/3)* * **b) sin (π/3 - 6π )** * = *sin(π/3 + 2π)* - **OR** - = *sin(-π/3)* * = *sin(π/3)* ### 3. Complementary Angles * **Determine the exact values of the six trigonometric ratios (use special triangles). Describe any relationships that you see. (π/3 and π/6 are complimentary angles. π/3 + π/6 = π/2 or 90°)** | Ratio | Value | |----------|----------------------| | sin(π/3) | √3/2 | | cos(π/3) | 1/2 | | tan(π/3) | √3 | | csc(π/3) | 2/√3 | | sec(π/3) | 2 | | cot(π/3) | 1/√3 | | Ratio | Value | |----------|-----------------------| | sin(π/6) | 1/2 | | cos(π/6) | √3/2 | | tan(π/6) | 1/√3 | | csc(π/6) | 2 | | sec(π/6) | 2/√3 | | cot(π/6) | √3 | * **Relationships:** * *sin(θ) = cos(π/2 - θ)* * *tan(θ) = cot(π/2 - θ)* * **Any right triangle, where θ is the measure of one of the acute angles, has a complementary angle of π/2 - θ for the other angle.** * **Find the complementary angle in the following triangle. ( 0 = π/8 and 3π/8 are complementary angles. 3π/8 + π/8 = π/2)** * **Complete the following table using the relationships you noticed above. Do not use a calculator.** | Ratio | Value | Complementary Angle | |-----------|------------|----------------------| | sin(3π/8) | 0.9239 | π/8 | | cos(3π/8) | 0.3827 | π/8 | | tan(3π/8) | 0.4142 | π/8 | | csc(3π/8) | 1/0.9239 | 4π/8 | | sec(3π/8) | 1/0.3827 | 4π/8 | | cot(3π/8) | 1/0.4142 | 4π/8 | ### 4. Related Acute Angles * **Revisiting the CAST rule:** * Given that θ is the acute angle from Quadrant I, it will be used as the related acute angle for the principal angles in Quadrants II, III and IV. * **Think:** Will the ratios be the same as they were in Quadrant I? If no, how will they be different? * **Complete the statements in each quadrant. Remember the CAST rule.** | Quadrant | sin(π - θ) | cos(π - θ) | tan(π - θ) | |----------|-----------|-----------|-----------| | Quadrant II | sin θ | - cos θ | - tan θ | | Quadrant | sin(π + θ) | cos(π + θ) | tan(π + θ) | |----------|-----------|-----------|-----------| | Quadrant III | - sin θ | - cos θ | tan θ | | Quadrant | sin(2π - θ) | cos(2π - θ) | tan(2π - θ) | |----------|-----------|-----------|-----------| | Quadrant IV | - sin θ | cos θ | - tan θ | * **Example 3** Use related acute angles to write an expression that is equivalent to the following expressions. * **a) tan(5π/3)** * **tan(5π/3) =-tan(π/3)** * **b) sin(5π/6)** * **sin(5π/6) = sin(π/6)** ### 7.2 Compound Angle Formulas * **compound angle** - an angle created by adding or subtracting 2 or more angles * you can use compound angle formulas to obtain **exact** values for trig ratios * you can use compound angle formulas to show equivalency | | Addition Formulas | Subtraction Formulas | |-----|--------------------|---------------------| | sin | sin(a + b) = sin a cos b + cos a sin b | sin(a - b) = sin a cos b - cos a sin b | | cos | cos(a + b) = cos a cos b - sin a sin b | cos(a - b) = cos a cos b + sin a sin b | | tan | tan(a + b) = (tan a + tan b) / (1 - tan a tan b) | tan(a - b) = (tan a - tan b) / (1 + tan a tan b) | **Example 1** Determine the exact value of each trigonometric ratio. * **a) sin(105°) = sin(60° + 45°)** * = *sin 60° cos 45° + cos 60° sin 45°* * = *(√3/2)(√2/2) + (1/2)(√2/2)* * = *(√6/4 + √2/4)* * = *(√6 + √2)/4* * **b) tan (12π/12) = tan(π + π/4)** * = *tan π + tan π/4* * = *(1 - tan π tan π/4)* * = *0 + 1* * = *1* **Example 2** * **If sin x = -4/5 and sin y = -12/13 and 0 < x < π/2 and 3π/2 < y < 2π , evaluate cos(x+y).** * *cos(x+y) = cos x cos y - sin x sin y* * = *(3/5)(5/13) - (-4/5)(-12/13)* * = *(15/65) - (48/65)* * = *-33/65* * **cos(x+y) = -33/65** ### 7.3 Double Angle Formulas * The double angle formulas show how trig ratios for double angles, 2θ, are related to trig ratios of the original angle, θ. | | Sine | Cosine | Tangent | |-----|------|--------|---------| | | sin 2θ = 2 sin θ cos θ | cos 2θ = cos²θ - sin²θ | tan 2θ = (2 tan θ) / (1 - tan²θ) | | | | cos 2θ = 2 cos²θ - 1 | | | | | cos 2θ = 1 - 2sin²θ | | **Example 1** * **Express (cos²π/8 - sin²π/8) as a single trigonometric ratio and then evaluate.** * = *(cos(2π/8))* * = *(cos(π/4))* * = *(√2/2)* * **Cos 2θ = cos²θ - sin²θ** **Example 2** * **Determine the values of sin2θ given cosθ = -4/5 and π/2 ≤ θ ≤ π.** * *sin 2θ = 2 sin θ cos θ* * *sin2θ = 2(3/5)(-4/5)* * **sin 2θ = -24/25** ### 7.4 Proving Trigonometric Identities * The following trigonometric identifies can be used to prove more complex trigonometric identities. | | Identities Based on Definitions | Identities Derived From Relationships | |-----|--------------------------|-----------------------------| | | **RECIPROCAL IDENTITIES** | **QUOTIENT IDENTITIES** | | | csc θ = 1/sin θ | tan θ = sin θ / cosθ | | | sec θ = 1/cos θ | cot θ = cos θ / sin θ | | | cot θ = 1/tan θ | | | | **PYTHAGOREAN IDENTITIES** | |------|-----------------------------| | | sin²θ + cos²θ = 1 | | | 1 + tan²θ = sec²θ | | | 1 + cot²θ = csc²θ | | | **DOUBLE ANGLE FORMULAS** | **ADDITION/SUBTRACTION FORMULAS** | |--------|----------------------------|-----------------------------------| | **SINE** | sin 2θ = 2 sin θ cos θ | sin(a + b) = sin a cos b + cos a sin b | | | | sin(a - b) = sin a cos b - cos a sin b | | **COSINE** | cos 2θ = cos²θ - sin²θ | cos(a + b) = cos a cos b - sin a sin b | | | cos 2θ = 2 cos²θ - 1 | cos(a - b) = cos a cos b + sin a sin b | | | cos 2θ = 1 - 2sin²θ | | | **TAN** | tan 2θ = (2 tan θ) / (1 - tan²θ) | tan(a + b) = (tan a + tan b) / (1 - tan a tan b) | | | | tan(a - b) = (tan a - tan b) / (1 + tan a tan b) | | | **EQUIVALENT TRIG FUNCTIONS** | |-----|--------------------------| | | **PERIODIC** | **SYMMETRY** | **HORIZONTAL TRANSLATIONS** | **COFUNCTION IDENTITIES** | | | sin θ = sin(θ + 2π) | cos θ = cos(-θ) | sin θ = cos(π/2 - θ) | sin θ = cos(π/2 - θ) | | | cos θ = cos(θ + 2π)| sin(-θ) = -sin(θ) | cos θ = sin(π/2 + θ) | cos θ = sin(π/2 - θ) | | | | tan(-θ) = -tan(θ) | | tan θ = cot(π/2 - θ) | **Example 1** Prove. * **a) cos(π/2 + x) = - sin x** * **L.S.** * = *cos π/2 cos x - sin π/2 sin x* * = *(0) cos x - (1) sin x* * = *-sinx* * **R.S.** * = *-sin x* * **b) cos(x - y) / cos(x + y) = (1 + tan x tan y) / (1 - tan x tan y)** * **L.S.** * = *(cos x cos y + sin x sin y) / (cos x cos y - sin x sin y)* * **R.S.** * = *(cos x cos y + sin x sin y) / (cos x cos y - sin x sin y)* * = *cos x cos y + sin x sin y* * = **cos x cos y + sin x sin y** * **c) tan 2x - 2 tan 2x sin²x = sin 2x** * *tan 2x (1 - 2 sin²x) = sin 2x* * *tan 2x (cos 2x) = sin 2x* * *(sin 2x cos2x) / (cos 2x) = sin2x* * **sin 2x = sin 2x** ### 7.5 Solving Linear Trigonometric Equations * Due to the *periodic* nature of trigonometric functions, trigonometric equations have an *infinite* number of solutions. For this reason, there is usually a specified interval for the solutions. **Example 1** * **How many solutions do the following functions have for any value of y when 0 ≤ x ≤ 2π ?** * **a) sin x** * sin x = 0 - 3 solutions * sin x = 1 - 1 solution * sin x = -1 - 1 solution * 0 < sin x < 1 - 2 solutions * -1 < sin x < 0 - 2 solutions * **b) cos 2x** * cos 2x = 0 - 4 solutions * cos 2x = 1 - 3 solutions * cos 2x = -1 - 2 solutions * 0 < cos 2x < 1 - 4 solutions * -1 < cos 2x < 0 - 4 solutions * **c) sin 4x** * sin 4x = 0 - 9 solutions * sin 4x = 1 - 4 solutions * sin 4x = -1 - 4 solutions * 0 < sin 4x < 1 - 8 solutions * -1 < sin 4x < 0 - 8 solutions **Example 2** * **Determine the solutions for the following equation, where 0 ≤ x ≤ 2π** * *-5 cos x + 3 = 2* * 1. Isolate the trig function * -5 cos x = - 1 * cos x = 1/5 * 2. cos is positive in QI and QIV * 3. Using a calculator, find the value of the related acute angle, B * cos B = 1/5 * B = cos⁻¹(1/5) = 1.37 rad * 4. Determine all other solutions using CAST/period * x₁ = 1.37 rad * x₂ = 2π - 1.37 = 4.91 rad **Example 3** * **Determine the solutions for the following equation on the interval 0 ≤ x ≤ 2π.** * *cos 2x = -1/2* * 1. k = 2, therefore the period is π * 2. cos is negative in QII and QIII * 3. cos B = 1/2 with a related acute angle, B = π/3 * 4. Determine all solutions using CAST/period * 2x₁ = π - π/3, 2x₂ = π + π/3 * x₁ = 2π/3, x₂ = 4π/3 * x₃ = 2π/3 + π, x₄ = 4π/3 + π * x₃ = 5π/3, x₄ = 7π/3 **Example 4** * **Solve 2 sin x cos x = cos 2x for the interval 0 ≤ x ≤ 2π.** * 1. *sin 2x = cos 2x* * 2. *tan 2x = 1* * 3. tan is positive in QI and QIII * 4. Using a calculator, find the related acute angle, B. Make sure your calculator is in radians. * tan B = 1 * B = tan⁻¹(1) = π/4 * 5. Determine all solutions using CAST/period * 2x₁ = B, 2x₂ = π + B * 2x₁ = π/4, 2x₂ = 5π/4 * 2x₃ = π/4 + π, 2x₄ = 5π/4 + π * x₁ = π/8, x₂ = 5π/8 * x₃ = 9π/8, x₄ = 13π/8 **Example 5** * **Today, the high tide in Matthews Cove, New Brunswick, occurs at midnight. The water level at high tide is 7.5m. The depth, *d* metres, of the water in the cove at time *t* hours is modeled by the equation *d(t) = 4 + 3.5cos(π/6 t)*. Jenny is planning a day trip to the cove tomorrow, but the water needs to be at least 2m deep for her to maneuver her sailboat safely. How can Jenny determine the times when it will be safe for her to sail into Matthews Cove?** * 1. Find *t*-values when d(t) = 2, for 24 hrs. * 2 = 4 + 3.5 cos(π/6 t) * -2 = 3.5 cos(π/6 t) * -0.5714 = cos(π/6 t ) * 2. Cos is negative in QII and QIII * B = cos⁻¹(0.5714) = 0.96 rad * 3. Determine the times when it will be safe for her to sail into matthews cove using CAST/period. * (π/6) t₁ = π - B * t₁ = (π - 0.96) * (6/π) = 2.18159 hrs or approximately 2:11 am * (π/6) t₂ = π + B * t₂ = (π + 0.96) * (6/π) = 4.10159 hrs or approximately 4:10 am * t₃ = t₁ + 12 = 14.18 hrs or approximately 2:11 pm * t₄ = t₂ + 12 = 16.16 hrs or approximately 4:10 pm ### 7.6 Solving Quadratic Trigonometric Equations * To solve a quadratic trigonometric equation: * Factor * Quadratic Formula * Use a Pythagorean Identity, compound angle formula or double angle formula to reduce the number of trigonometric functions in the equation. **Example 1** * **Solve each equation for x in the interval 0 ≤ x ≤ 2π.** * **a) sin²x - sin x = 2** * sin²x - sin x - 2 = 0 * Let m = sin x * m² - m - 2 = 0 * (m + 1)(m - 2) = 0 * (sin x + 1)(sin x - 2) = 0 * sin x = - 1 or sin x = 2 * sin x = -1 has solutions but sin x =2 has no solutions. The only solution to the trigonometric equation is x =3π/2 * ** b) 2sin²x - 3sin x + 1 = 0** * let z = sin x * 2z² - 3z + 1 = 0 * 2z² - 2z - z + 1 = 0 * 2z(z - 1) - (z - 1) = 0 * (2z - 1)(z - 1) = 0 * (2 sin x - 1) (sin x - 1) = 0 * 2 sin x - 1 = 0 or sin x - 1 = 0 * sin x = 1/2 or sin x = 1 * x₁ = π/6 or x₃ = π/2 * x₂ = 5π/6 or x₄ = 3π/2 **Example 2** * **For each equation, use a trigonometric identity to create a quadratic equation. Then solve the equation for x in the interval [0, 2π].** * **3sinx + 3cos2x = 2** * 3sinx + 3 (1 - 2sin²x) = 2 * 3sinx + 3 - 6sin²x - 2 = 0 * -6sin²x + 3sinx + 1 = 0 * Using the quadratic formula: * sinx = [-b ± √(b² - 4ac)] / 2a * sin x = [-3 ± √(3² - 4(-6)(1))] / 2(-6) * sin x = [-3 ± √(33)] / -12 * sin x = (3 ± √(33)) / 12 * sin x = (3 + √(33)) / 12 = 0.7287 * x₁ = sin⁻¹(0.7287) = 0.82 rad * x₂ = π - sin⁻¹(0.7287) ≈ 2.32 rad * sin x = (3 - √(33)) / 12 = -0.2287 * x₃ = π + sin⁻¹(0.2287) ≈ 3.37 rad * x₄ = 2π - sin⁻¹(0.2287) ≈ 6.05 rad