Table of Contents 1. Angles pg. 2 and 32. DMS and DD pg. 43. Reference Angles pg. 54. Sine, Cosine, Tangent pg. 6 and 75. Reciprocal Functions pg. 86.
The Unit Circle and Quadrant Angles pg. 97. Unit Circle pg. 108. Using TI-82 and 83's to find Trig. Function Values pg. 119. Solving Right Triangle's pg. 1210. Law of Sines and Law of Cosines and The Ambiguous Case pg. 1311.
Trigonometric Ratios pg. 14 and 1512. Area of Triangles pg. 1613. Graphing Trig. Functions on the TI-83 plus pg. 1714. Phase Displacement pg. 1815. Graphing Cosine and Sine trig. fun.
By hand pg. 19 and 2016 Graphing Tangent and Cot trig. Func. By hand pg. 21 and 2217. Self Evaluation pg. 23 Angles Angles can be measured in three ways: - Degrees- Radians- Revolutions (or rotations) 1 Revolution = 2π radians = 360 degrees positive angle rotates in a counter clockwise direction.
A negative angle rotates in a clockwise direction. Revolutions: 1/4 CCW = 2 1/2 CW = 3/4 rot CW = 1, 3, 4, 7, and 9 are co terminal angles. Circumference of a circle formula is C = 2 π R, therefore the central angle of 1 full revolution CCW corresponds to an arc length of S = 2 π R. Central angle is measured in radians and has the formula = S / R or S = R If 360 degrees = 2 π radians, then: 360/2π = 1 radian or 180/π . To convert degrees to radians: multiply by π /180. To change radians to degrees: multiply by 180/π . Change 30 degrees to radians: 30/π π /180 = π /6 radians.
Change 3/4π radians to degrees: 3/4π 180/π = 135 degrees. 310 degrees = 31/18π 3/2π = 270 degrees. DMS and DD 1 degree contains 60 minutes. 1 minute contains 60 seconds.
- DMS means degrees, minutes, seconds- DD means decimal degrees. - To convert DD to DMS just enter the DD value, hit 2nd angle, choice 4 (DMS), enter, and you " ll get DMS. - To convert DMS back to DD on a TI-82, enter as follows: 10 degrees 15'30" will be entered as: 10 + 15/60 + 30/3600 = and you " ll get the DD result. - To convert radians to degrees you first convert the radians, then you have DD. If you need DMS you can use your calculator, as stated above.
If is in Q 1, then is its own ref. angle. If is in Q 2, then 180 degrees - is the ref. angle. If is in Q 2, then π - is the ref. angle. If is in Q 3, then -180 degrees is the ref. angle. If is in Q 3, then - π is the ref. angle.
If is in Q 4, then 360 degrees - is the ref. angle. If is in Q 4, then 2π - is the ref. angle. Reference angles are always positive and less than 90 degrees. Sine, Cosine, Tangent SinA = opposite / hypotenuse = 4/5 Cos = adjacent / hypotenuse = 3/5 TanA = opp. /adj.
= 4/3 SOHCAHTOASinB = 3/5 CosB = 4/5 TanB = 3/4 Sin is the reciprocal of the cosecant (CSC). CSC = hyp. /opp. Secant (Sec) Sec = hyp. /adj. = 5/3 (CosA) (SecA) = 1 Cotangent (Cot) = CotA = adj. /opp. = 3/4 Tan = 1/Cot Cot = 1/Tan (Tan) (Cot) = 1 (Sin) (CSC) = 1 Sin 120 degrees = Sin of 60 degrees angleR = Hypotenuse Sin = y / r Sec = r / x Cos = x / r Csc = r / y Tan = y / x Cot = x / y P is a point on the terminal side of an angle in standard position.
When you construct a from any point P on the terminal side to the horizontal axis, a right triangle is formed. This triangle is called the Reference Triangle. The distance r from the origin to point P on the terminal side of an angle in standard pos. is the radius vector. You can find the radius vector r by using- Pythagorean Theorem. R = x 2 + y 2, ... distance is always positive Special ratios: 1. Sine = y / r 2.
Cosine = x / r 3. Tangent = y / x 4. CSC = r / y 5. Sec = r / x 6.
Cot = x / y EX: Given the value of one trig. Ratio, you can find the values of the other five; . For ex. Tan. Then find Sec, Csc, and Cot.
Sin = 2/7 Csc = 7/2 Cos = -3 sq. rt. Of 5/7 Sec = -7 sq. rt. Of 5/15 Tan = -2 sq. rt. Of 5/15 Cot = -3 sq. rt. Of 5/2 Reciprocal Functions Sin - csc Cos sec Tan cot Sin X csc = 1 Cos X sec = 1 Tan X cot = 1 Because sin and csc are reciprocals: csc = 1/sin sin isn't equal to 0, ... same goes for cos and sec, ... and tan and cot. Sec 0 and Csc 0 are in Q 1 The Unit Circle and Quadrant Angles- x 2 + y 2 is the equation for the unit circle (radius = 1) - Sine = y / r = y/1 = y}- Cos = x / r = x/1 = x } P (x, y) The ordered pair for pt.
P is on the circle and represents (Cos, Sine) By Choosing a convenient value for r, you can also evaluate the trig. Function values for the "special" angles 30 deg., 60 deg., and 45 deg. In geometry, ... 30 deg. - 60 deg. - rt. Triangles have sides in the ratio of: 1: 2 sq. rt.
Of 2 Gie low says: How does this effect the values for the trig functions in the unit circle (Sin 2) 45 deg. Cos 60 deg. - Tan 60 deg. Function Values Set the mode on the calculator to radian or degree as needed. Degree mode: with TI -82's, you press the fun. Key, then the value, then enter: Sin 30 then hit enter and you should get.
5. Examples: Cos 81 deg. = . 9093 Csc 2 = 1/sin 2 enter = 1.0998 Given the trig function value, you can find. Ex: Csc = 1.5557, ... do the following: 2nd Sin -1 (1 divided by 1.5557) = 40 deg. Ex 2: Cos = -.
7071 = 2.35618 ( ) Given two values of (0 360 deg.) Sin = 1/2, ... 30 and 150 Sine Function values increase from 0 deg. To 90 deg. Solving Right Triangle " sTo solve a right triangle means to find all of the measurements of its sides and angles. Use the trig. functions to help solve right triangles. 1.
Solve right triangle ABC. Round angle measures to nearest degree and side measures to the nearest tenth. Ang. A = 49 deg. Side a = 7 Ang B = 41 deg.
Side b = 6.1 Ang C = 90 deg. Side c = 9.3 Surveyors use trig. concepts and angle concepts often. An angle of elevation is the angle between a horizontal line and the line of sight from the observer to the object at a higher level. An angle of depression is the angle between a horizontal line and the line of sight from the observer to the object at a lower level. Oblique triangles are not right triangles.
Use the law of sines or the law of cosines to solve the right triangle. Law of Sines and Law of Cosines and The Ambiguous Case Law of Sines: a / SinA = b / SinB = c / SinC or SinA / a = SinB / b = SinC / cLaw of Cosines: a 2 = b 2 + c 2 - 2 bc CosA b 2 = a 2 + c 2 - 2 accost c 2 = a 2 + b 2 - 2 abCosCDetermine which law to use based on: 1.2 angles and 1 side = AAS or ASA = Law of Sines 2.2 sides and 1 angle (opposite one of these sides) = SSA (this is the ambiguous case) = law of sines 3.2 sides and the included angle = SAS = Law of Cosines 4.3 sides = = Law of cosines The ambiguous case. (SSA), When given 2 sides and 1 angle opposite one of these given sides you know to use the law of sines. But for this type of problem you have an ambiguous case.
To determine if you have 1 solution or two, test: If side c is than the other given side, then there are 2 solutions If side c is than the other given side, then there is only 1 solution. Use law of sines. Ex: Solve ABC if a = 12, b = 8, and B = 40 deg. (two solutions because side b side a) Trigonometric Ratios To do trig ratios use this simple mnemonic to remember the following ratios: Oscar Has A Heap Of Apples. Leg opposite to Hypotenuse Leg adjacent to X 1. Sine x = Opposite leg Hypotenuse 2. Cosine x = Adjacent leg Hypotenuse 3.
Tangent x = Opposite Adjacent lego These trigonometric ratios hold only for right triangles. Ex. 5 4 3 sin A = 4/5 sinB = 3/5 cos A = 3/5 cosB = 4/5 tanA = 4/3 tanB = 3/4 Area of Triangles You are already familiar with the formula A = 1/2 bh. Other formulas are also available to use and they are: 1. Hero's formula (or heron's formula): SSA = (The square root of) S (S-A) (S-B) (S-C) When S = 1/2 (A+B+C) This formula is used when given the lengths of the 3 sides of any triangle 2. K = 1/2 abSinCK = 1/2 acSinBK = 1/2 bcSinAUsed for any triangle when given 2 sides and the included angle. (note: all 3 of these formulas have the same pattern: 1/2 x one side x other side x sine of the included angle) 3.
If given any 2 angles and 1 side then use one of the following: K = 1/2 a (squared) x SinB SinC divided by Sina = 1/2 b (squared) x SinA SinC divided by Sinb = 1/2 c (squared) x SinA SinB divided by Since. Using the the 2nd method SAS. Angle A = 43 degrees side b = 16, side c = 12 1/2 (16) (12) Sin 43 = 65.5 sq. units Graphing Trig. Functions on The TI-83 plus Be sure the calculator is in Radian Mode. Clear out all old graphs.
Set the window to: X-min = - π X-max = 2 π X's cl = 1 y-min = -6 Y-max = 6 Y's cl = 1 Ex. Plug in: Y = sin (x) You get: Phase Displacement = (+ or -) sinb (x (+ or -) c) (+ or -) d 2 methods are commonly used in graphing phase displacement. The basic characteristics for sine graphs are still used to help you graph. (5 points to locate, 3 zeros + a maximum and a minimum use the a. o. 's. to put the "zeros" (1st, 3rd, and 5th tick marks on). The two methods are: 1. Graph the line (use dashes and do it lightly) first then shift it horizontally according to the phase displacement.
Erase the 1st "guiding" line. 2. Set up 2 equations which will determine the beginning and the end of the cycle. bx (+ or -) bc = 0 and bx (+ or -) bc = 2 π Solve for "x". The two results will be the starting location and ending location of one full cycle or interval of the equation with its phase shift. Graphing Cosine and Sine Trig. functions by hand First off here are a few definitions to help: 1. Amplitude- Amplitude is the distance the curve goes above or below the axis of symmetry.
2. Axis of symmetry- The AOS is a horizontal line that goes through the middle of the "wave"3. Period- the period is how far the wave goes before it begins to repeat its pattern. For Cosine and Sine you use these 6 steps: 1. (+ or -) gives you reflection 2. |a| gives amplitude 3. (+ or -) d gives the axis of symmetry. Phase Displacement Sine: Y = (+ or -) aSin bx (+ or -) d If it's + reflection it goes up to start with therefore it's not reflected, if it's a - reflection it goes down to start with therefore it's reflected. Always looks like a hump on a camels back, like this: Cosine: Y = (+ or -) cosb (x (+ or -) c) (+ or -) d If the cosine function is not reflected it goes down to start with, if it is reflected it goes up.
Ex. When reflected: 1st and 5th = max 2nd and 4th = a. o. 's. 3rd = min When not reflected: 1st and 5th = min 2nd and 4th = a. o. 's. 3rd = max (+ or -) c gives phase displacement+c to the left-c to the right Graphing tangent and cotangent functions by hand Unlike Sine and Cosine fnc's. the tangent fun. has "breaks" in its graph.
Also the basic Pd. Is π rather than 2 π like cosine and sine. Tangent functions: Notice: The tangent period starts and stops with zr eos and has an asymptote in the middle. If not refl. - from the 1st zero you increase toward the asymptote and from the 2nd zero you go down toward the asymptote. If refl. - from the 1st zero you go down toward the asymptote and from the 2nd zero you go up toward the asymptote. Think of the tangent function as in a design like: | The tangent function increases from left to right.
Y = (+ or -) a TanB (x (+ or -) c) (+ or -) cotangent: Y = cot (x) We get the basic cotangent by taking the reciprocal of the tangent. If tan increases then cot decreases. The reciprocal of 0 is "undefined" if an asymptote. The basic period like in tangent is π . The asymptotes are at the beginning and end of the period, the zero is in the middle. The cot.
Curve falls from left to right. Y = (+ or -) |a| Cot (x (+ or -) c) ) + or -) d Note: Think of cotangent as: | |.