Pre-Calculus Q2 Review PDF

Pre-calculus Review 2nd Quarter Periodical Test Review WHAT IS IT? ▪ Comes from two Greek words, trigonon (triangle) and metria (measure) ▪ A branch of mathematics, specifically of geometry, that studies measurement of triangles ...

Pre-calculus Review 2nd Quarter Periodical Test Review WHAT IS IT? ▪ Comes from two Greek words, trigonon (triangle) and metria (measure) ▪ A branch of mathematics, specifically of geometry, that studies measurement of triangles TRIGONOMETRIC FUNCTIONS QII QI 𝒚 sin 𝜽 = 𝑷 𝒙, 𝒚 𝒓 Radius 𝒓 𝒚 𝒙 cos 𝜽 = 𝒓 𝒙 𝒚 tan 𝜽 = 𝒙 QIII QIV TRIGONOMETRIC FUNCTIONS The other three trigonometric functions – cosecant, secant, and cotangent – are given by: 1 𝑟 csc 𝜃 = = sin 𝜃 𝑦 𝒚 1 𝑟 sec 𝜃 = = cos 𝜃 𝑥 1 𝑥 cot 𝜃 = = tan 𝜃 𝑦 TRIGONOMETRIC FUNCTIONS = 𝑷(cos 𝜽, sin 𝜽) = sin 𝜽 = cos 𝜽 #SagutinMoNaAko Trigonometry ▪ Suppose 𝜃 is an angle in standard position and let 𝑃(𝜃) = 𝑃(𝑥, 𝑦) be the point on its terminal side on the unit circle. Which trigonometric function is equal to 𝑦? __________ ▪ Suppose 𝜃 is an angle in standard position and let 𝑄(𝜃) = 𝑄(𝑥, 𝑦) be the point on its terminal side on a circle with radius 𝒓. What is cos 𝜃? __________ SOME COMMON ANGLES ANGLE MEASURE ▪ A degree, usually denoted by the symbol °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. 𝟒𝟓° 𝟑𝟔𝟎° ANGLE MEASURE ▪ An alternative measurement is the radian: One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. ANGLE MEASURE ▪ We can convert between radians and degrees via 𝝅 𝟔 rad 1. 30° = ____ 5𝜋 2. rad = ____° 𝟏𝟓𝟎 6 #SagutinMoNaAko Trigonometry ▪ What is the radius of the unit circle? _____ ▪ How many degrees are there in one revolution? _____ ▪ How many radians are there in one revolution? _____ ▪ Convert the following to radian measure: ❑ 135° = _____ rad ❑ 210° = _____ rad ▪ Convert the following to degree measure: ❑ 3𝜋/4 = _____° ❑ 5𝜋/6 = _____° TRIGONOMETRY An angle is formed by rotating a ray about its endpoint. It is said to be in standard position if the vertex of the angle is at (0,0) and the initial side of the angle lies along the positive x-axis. TRIGONOMETRY An angle measure is positive An angle measure is negative if the ray of the angle rotates if the ray of the angle rotates counterclockwise from the clockwise from the initial side initial side to terminal side. to terminal side. TRIGONOMETRY Coterminal angles are angles in standard position that have a common terminal side. TRIGONOMETRY Let 𝜃 be a non-quadrantal angle in standard position. The reference angle of 𝜃 is the acute angle 𝜃𝑅 that the terminal side of 𝜃 makes with the 𝑥-axis. TRIGONOMETRY Let 𝜃 be a non-quadrantal angle in standard position. The reference angle of 𝜃 is the acute angle 𝜃𝑅 that the terminal side of 𝜃 makes with the 𝑥-axis. #SagutinMoNaAko Trigonometry ▪ What do you call an angle whose vertex is at the origin and initial side is in the positive x-axis? ___________________ ▪ Which angle between −180° and 0° is coterminal with 330°? ▪ Which of the following is a possible measure of the angle illustrated below? A. −40° B. 40° C. 400° D. −400° #SagutinMoNaAko Trigonometry ▪ What is the reference angle of 330°? _____ ▪ The reference angle of 420° is 60°. What is the exact value of cos 420°? _____ ▪ What is the value of cos 225°? _____ ▪ Which of the following is equal to cos 225°? _____ A. − cos 25° B. − cos 45° C. cos 45° D. sin 45° TRIGONOMETRY Trigonometric Domain Range Function 𝑦 = cos 𝑥 ℝ or −∞, ∞ −1, 1 𝑦 = sin 𝑥 ℝ or −∞, ∞ −1, 1 TRIGONOMETRY Terms and Definitions: ❑ The amplitude is the height from the center line to the peak (or to the trough). Or we can measure the height from highest to lowest points and divide that by 2. ❑ The period goes from one peak to the next (or from any point to the next matching point). ❑ The horizontal shift, also called phase shift, is how far the function is shifted horizontally from the usual position. ❑ The vertical shift is how far the function is shifted vertically from the usual position. TRIGONOMETRY 𝑦 = 𝒂 sin 𝒃 𝑥 − 𝒄 + 𝒅 𝑦 = 𝒂 cos 𝒃 𝑥 − 𝒄 + 𝒅 where ▪ amplitude = 𝑎 2𝜋 ▪ period = 𝑏 ▪ horizontal shift = 𝑐 (to the right if 𝑐 is positive, to the left if 𝑐 is negative) ▪ vertical shift = 𝑑 (upward if 𝑑 is positive, downward if 𝑑 is negative) TRIGONOMETRY 𝑦 = 𝒂 sin 𝒃 𝑥 − 𝒄 + 𝒅 𝑦 = 𝒂 cos 𝒃 𝑥 − 𝒄 + 𝒅 Example: Identify the values of 𝑎, 𝑏, 𝑐, 𝑑, amplitude, period, horizontal 𝜋 and vertical shifts of 𝑦 = −2 sin 3 𝑥 − + 5. 4 𝜋 Solution: 𝑎 = −2, 𝑏 = 3, 𝑐 = , 𝑑 = 5 4 2𝜋 2𝜋 Amplitude = 𝑎 = −2 = 2, Period = = 𝑏 3 𝜋 Horizontal shift = units to the right, Vertical shift = 5 units upward 4 #SagutinMoNaAko: Identify the values of 𝑎, 𝑏, 𝑐, 𝑑, amplitude, period, horizontal and vertical shifts of 𝑦 = −5 cos 3 𝑥 − 𝜋 + 2. TRIGONOMETRY Recall from General Mathematics: The relation reversing the process done by any function 𝑓 𝑥 is called the inverse of 𝒇 𝒙 (or in symbol, 𝒇−𝟏 𝒙 ). Also, the inverse of a function exists if the function is one-to-one function. Hence, given that the function is a one-to-one, the domain of the inverse is the range of the original function, and the range of the inverse is the domain of the original function. Although all trigonometric functions are not one-to-one, we can still find its domain and range by restricting their respective domains. We then define each respective inverse function and illustrate the domain and range of trigonometric functions. TRIGONOMETRY Inverse Trigonometric Domain Range Function 𝑦 = cos −1 𝑥 or 𝑦 = arccos 𝑥 −1, 1 0, 𝜋 𝜋 𝜋 𝑦 = sin−1 𝑥 or 𝑦 = arcsin 𝑥 −1, 1 − , 2 2 TRIGONOMETRY Remember: When asked for inverse trigonometric function values, we are now looking for angles (within the range) with the given trigonometric value. Also, arcsin 𝜃 is the same as sin−1 𝜃 and arccos 𝜃 is the same as cos −1 𝜃. Examples: Find the exact value of each expression. −1 1 𝝅 𝜋 1 1. sin = 𝟑𝟎° or since sin 30° or sin = 2 𝟔 6 2 𝝅 2. arcsin −1 = −𝟗𝟎° or − since sin(−90°) = −1 𝟐 −1 𝝅 3. cos 0 = 𝟗𝟎° or since cos 90° = 0 𝟐 𝝅 4. arccos cos 270° = arccos 0 = 𝟗𝟎° or 𝟐 #SagutinMoNaAko Trigonometry ▪ What is the domain of 𝑦 = arccos 𝑥? _____ ▪ What is the range of 𝑦 = sin−1 𝑥? _____ −1 2 ▪ What is the exact value of sin ? _____ 2 −1 1 ▪ What is the exact value of cos ? _____ 2 ▪ What is the exact value of arccos(cos 2𝜋)? _____  Terms and Definitions TRIGONOMETRY ❑ The domain of an expression (or equation) is the set of all real values of the variable for which every term (or part) of the expression (or equation) is defined in ℝ. ❑ An identity is an equation that is true for all values of the variable in the domain of the equation. ❑ An equation that is NOT an identity is a conditional equation. ❑ If some values of the variable in the domain of the equation do not satisfy the equation, then the equation is a conditional equation. ❑ A trigonometric equation is an equation that involves any trigonometric expressions. ❑ A trigonometric identity is an equation involving any of the six trigonometric functions that holds for any real number for which each member of the equation is defined. TRIGONOMETRIC IDENTITIES RECIPROCAL IDENTITIES 1 1 1 csc 𝜃 = sec 𝜃 = cot 𝜃 = sin 𝜃 cos 𝜃 tan 𝜃 QUOTIENT IDENTITIES sin 𝜃 cos 𝜃 tan 𝜃 = cot 𝜃 = cos 𝜃 sin 𝜃 PYTHAGOREAN IDENTITIES sin2 𝜃 + cos 2 𝜃 = 1 tan2 𝜃 + 1 = sec 2 𝜃 1 + cot 2 𝜃 = csc 2 𝜃 TRIGONOMETRIC IDENTITIES EVEN-ODD IDENTITIES sin −𝜃 = − sin 𝜃 cos −𝜃 = cos 𝜃 tan −𝜃 = −tan 𝜃 COFUNCTION IDENTITIES cos 90° − 𝐵 = sin 𝐵 sin 90° − 𝐵 = cos 𝐵 tan 90° − 𝐵 = cot 𝐵 TRIGONOMETRY Tips in solving trigonometric equations: 1. If the equation contains only one trigonometric term, isolate that term, and solve for the variable. 2. If the equation is quadratic in form, we may use factoring, finding the square roots, or the quadratic formula. 3. Rewrite the equation to have 0 on one side and then factor (if appropriate) the expression on the other side. 4. If the equation contains mote that one trigonometric function, try to express everything in terms of one trigonometric function. Here, identities are useful. TRIGONOMETRY Example 1: Solve the equation 2 cos 𝑥 − 1 = 0. Solution: Adding 1 to both sides of the equation, 2 cos 𝑥 − 1 + 𝟏 = 0 + 𝟏 → 2 cos 𝑥 = 1 Dividing both sides by 2, 2 cos 𝑥 1 1 = → cos 𝑥 = 𝟐 𝟐 2 𝜋 5𝜋 The angles whose cosine value is ½ are and and all their 3 3 coterminal angles. 𝝅 The complete solutions of the equation are 𝑥 = + 𝒌(𝟐𝝅) and 𝟑 𝟓𝝅 𝑥= + 𝒌(𝟐𝝅) for all integers 𝑘. 𝟑 TRIGONOMETRY Example 2: Determine all solutions of 4 cos2 𝑥 − 1 = 0 within the interval 0, 𝜋. Solution: Adding 1 to both sides of the equation, 4 cos2 𝑥 − 1 + 𝟏 = 0 + 𝟏 → 4 cos2 𝑥 = 1 Dividing both sides by 4, 4 cos2 𝑥 1 2 1 = → cos 𝑥 = 𝟒 𝟒 4 Getting the square root of both sides, 1 cos 𝑥 = 2 The only angle within 0, 𝜋 whose cosine value is ½ is 𝟔𝟎°. #SagutinMoNaAko Trigonometry ▪ What do you call an equation that is true for all values of the variable in the domain of the equation? ________________ ▪ What do you call an equation that involves any trigonometric expressions? ________________ tan 𝜃 cos 𝜃 ▪ Use the tangent quotient identity to simplify. sin 𝜃 #SagutinMoNaAko Trigonometry ▪ Determine whether the given is an identity or not. 1. 1 − cos 2 𝜃 = sin2 𝜃 2. 1 + tan2 𝜃 = sec 2 𝜃 3. sec 𝜃 = cos 90° − 𝜃 4. cos 𝜃 = sin 90° − 𝜃 ▪ Determine all solutions of 4 cos 2 𝑥 − 3 = 0 within the interval 0, 𝜋.