Trigonometry Review. Table of Contents
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1 Jesuit High School Math Department Table of Contents Subject Page Worksheet. Angles and their measure. Angles and their measure Degrees. Angles and their measure Radians.4 Angles and their measure Arc Length.5 Angles and their measure Sector Area 4. Unit circle approach 5. Familiar Angles 6. Co terminal angles 7.4 Examples from Worksheet 8.5 Unit Circle Diagram 0. Properties of trig functions. Signs of trig functions 4 4. Trig Equations 6 4, 5 5. Not Available 6. Sine Rule Sine Rule Ambiguous Cases Law of Cosines 7 Rev 5//07, T. Lane
2 Section. Angles and Their Measure Angles can be denoted by lower case Greek letters, such as α (alpha), β (beta), γ (gamma), and θ (theta) Standard position vertex at origin and Counter clockwise positive direction initial side on positive x axis. Example: Θ 0º Clockwise negative direction Example: Θ 0º Quadrants when an angle lies on the x or y axis rather than in a quadrant, we say the angle is a quadrantal angle. Quadrant Diagram Section. Angles and Their Measure Degrees Degrees: 60 complete revolution Right angle 90 ¼ revolution 5% of 60 Example : draw each angle a) 50 c) 00 b) 80 d) 90 (60 + 0) of
3 Section. Degrees, minutes, seconds: minute (/60) degree 60 minutes degree 60 Example : Convert a).464 to D/M/S + 60(0.464) (.84) Angles and Their Measure Degrees second (/60) minute (/600) degree 600 seconds 60 minute degree On Calculator, the keystrokes are: [.464] [ nd ] [Angle] [4] b) 4 45 to degrees + (4/60) + (45/600).75 Problems: Worksheet :, Section. Angles and Their Measure Radians The circumference of a circle is an arc length. The ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of π. That ratio of the circumference (C) of a circle to the radius (r) is π, and is the radian measure of revolution. C circumference D diameter r radius s arc length Definition of π: π C D C r The ratio of any arc length to the radius, or s/r, will be the radian measure of the central angle which that arc subtends. revolution: C π r Measure of a central angle: The radian measure is a real number that indicates the s ratio of a curved line to a straight line, or an arc to the Θ radius. r Convert DEG to RAD: multiply by π/80 π radians 80 Convert RAD to DEG: multiply by 80/π radian (80/π) 57.9 π/80 radians r radius s arc length Θ central angle If Θ radian, then s radian of
4 Section. Angles and Their Measure Radians Example : convert to degrees π π 80 ( 80 ) 6 π 6 π 80 6 ( 80 ) a) radians 7 b) radians π π 5 80 c). 7 radians. 7 rad. 99 π Example : convert to radians π π π 5 π a) radians b) radians π 5 π c) radians Example : sketch the angle 8π 8 ( 80 ) 480 Problems: Worksheet : 6, Section.4 Angles and Their Measure Arc Length Arc Length s rθ Where r radius And θ central angle measured in RADIANS Examples: find the arc length, given: r 5 cm a) central angle π/ radians s rθ 5(π/) 5.6 cm b) central angle 0.5 radians s rθ 5(0.5).5 cm c) central angle 0 s rθ 5(0)(π/80) 5π/6.68 cm Also, the ratio of arc length (s) to circumference (C), or the whole circle, is equal to the ratio of the central angle (θ) to 60, or the whole circle. π 60 Problems: Worksheet : 7 44 s r Θ of
5 Section.5 Angles and Their Measure Sector Area Derivation: Area of a Sector r Θ area of sec tor area of circle A r Θ π r π r length of arc circumfere nce r Θ π r π r ( ) and A r Θ Also, the ratio of sector area (A S ) to area of the circle (A C ), or the whole r Θ circle, is equal to the ratio of the central Θ angle (θ) to 60, or the whole circle. π r 60 Example: find the area of a sector in a circle with r 5 cm and θ 45 π A r Θ (5 )( 45 ) 9. 8 cm 80 Problems: Worksheet : 45 56, 8 84 When working problems with a calculator, ensure it s in the correct mode DEGREES or RADIANS To change the mode, press MODE and move the cursor to the appropriate choice, then press ENTER. Examples: Degree mode Sin 4º Cos 70º 0.4 Cot 50º /tan radian mode Examples: Tan 5π/.7 Sec π/ cos π 4 of
6 Section. Unit Circle Approach Trigonometric Functions: sin θ opp/hyp csc θ /sin θ hyp/opp cos θ adj/hyp sec θ /cos θ hyp/adj tan θ opp/adj cot θ /tan θ adj/opp Unit Circle has radius Positive angles are measured counterclockwise from the positive x axis. Point P is a point on the unit circle with coordinates (x,y) y sin t y x cos t x y tan t x csc t y sec t x x cot t y Note: if x 0, i.e., it is a point on the y axis (0,y), then sec t and tan t are undefined. Also, if y 0, i.e., it is a point on the x axis (x,0), then cosec t and cot t are undefined. Example: P 5,. Find all 6 trig function values. sin t cos t 5 (y coordinate) (x coordinate) y 5 tan t x Problems: Worksheet : 8, 7 8 csc t y sec t x x cot t tan t y 5 5 of
7 Section. Familiar Angles 0, 60, and 45 sin 0 cos 0 tan 0 csc 0 sec 0 cot Triangle: sin 60 cos 60 tan 60 csc 60 sec 60 cot 60 Equilateral triangle: By Pythagoras Thm: + h and h sin 45 cos 45 tan 45 csc 45 sec 45 cot 45 Square: By Pythagoras Thm: x + and x Key angles in degrees and radians: 0 π radians 6 60 π radians 45 π radians 4 6 of
8 Section. Co terminal Angles Co terminal angles have the same initial and terminal sides. To find all angles that are co terminal with 40, add or subtract multiples of 60, or (where n integer): n To find all angles that are co terminal with π, add or subtract multiples of π, or (where n integer): π + πn Example: (, 5) is a point on the terminal side of angle θ. Find the exact values of the 6 trig functions of θ. sin θ 5/ csc θ /5 cos θ / sec θ / tan θ 5/ cot θ /5 Example: (, ) is a point on the terminal side of angle θ. Find the exact values of the 6 trig functions of θ. r sin Θ 5 cos ec Θ cos Θ sec Θ 5 tan Θ cot Θ / r + 5 Problems: Worksheet : of
9 Section.4 Examples from Worksheet #, #0 cos 7 π The point, sin t cos t, sin t, therefore: csc t cos t sec t tan t cot t 7π 6π + π ()()π + π where n. Therefore 7π and π are coterminal and cos 7 π cos π #0 sin 0 cos 45 #0 #4 #8 sin 4 π tan 4 sin 0 cos 45 π + sin 45 + tan 45 ( ) ( ) + π π csc + cos 6 csc 60 + cos π sec π csc cos π sin π cos 80 sin 90 cos 80 sin 90 Problems: Worksheet : of
10 Section.4 Examples from Worksheet #40 5π 6 #4 40 #44 50 The associated acute angle of 50 is 0 The associated acute angle of 40 is 60 π (co terminal) The associated acute angle of 5 is 45 #54 5π (co terminal) sin 50 / cos 50 tan 50 cosec 50º sec 50 cot 50 sin 40 cos 40 tan 40 cos ec 50 sec 40 tan 40 Problems: Worksheet : 9 56, 8 90 sin 495 cos 495 tan 495 cos ec 495 sec 495 cot 495 Sin θ 0 Cos θ Tan θ sinθ/cosθ 0 Cosec θ /0 (undefined) Sec θ Cot θ /0 (undefined) P (,0) (cosθ, sinθ) 9 of
11 Section.5 Unit Circle Diagram 0 of
12 Section Properties of Trig Functions Properties of the Sine function:. the domain is the set of all real numbers. the range consists of all real numbers from to, inclusive. the sine function repeats itself every 60 or π. 4. the x intercepts are..., π, π, 0, π, π, π, the y intercept is 0 6. the maximum value is and occurs at x..., π/, π/, 5π/, 9π/ 7. the minimum value is and occurs at x..., π/, π/, 7π/, π/ x Sin x y sin x sinθ sin(θ + 60 k) or sin(θ + πk) where k is an integer. Properties of the Cosine function:. the domain is the set of all real numbers. the range consists of all real numbers from to, inclusive. the cosine function repeats itself every 60 or π. 4. the x intercepts are..., π/, π/, π/, π/, 5π/, the y intecept is 6. the maximum value is and occurs at x..., π, 0, π, 4π, 6π 7. the minimum value is and occurs at x..., π, π, π, 5π y cos x cos θ cos(θ + 60 k) or cos(θ + πk) where k is an integer. of
13 Section Properties of Trig Functions Properties of the Tangent function:. the domain is the set of all real numbers, except odd multiples of π/.. the range consists of all real numbers ( to ). the tangent function repeats itself every 80 or π. 4. the x intercepts are..., π, π, 0, π, π, π, the y intercept is 0 6. the vertical asymptotes occur at x..., π/, π/, 5π/, 9π/ y tan x tanθ tan(θ + 80 k) or tan(θ + πk) where k is an integer. Properties of the Cosecant function:. csc x sin x. undefined when sin x 0. domain the set of all real numbers, except 0 and multiples of π. 4. the period π 5. no y intercept 6. the vertical asymptotes occur at x..., π, 0, π, π,... nπ 7. range: csc, csc x,, x, ( ) ( ) y csc x cscθ csc(θ + 60 k) or csc(θ + πk) where k is an integer. of
14 Section Properties of Trig Functions Properties of the Secant function:. sec x cos x. undefined when cos x 0. domain the set of all real numbers, except multiples of π/. 4. the period π 5. y intercept 6. the vertical asymptotes occur at x..., π/, π/, π/,... nπ/, where n odd integers 7. range: sec, sec x,, x, ( ) ( ) y sec x sec θ sec(θ + 60 k) or sec(θ + πk) where k is an integer. Properties of the Cotangent function:. cot x tan x. undefined when tan x 0. domain the set of all real numbers, except 0 and multiples of π. 4. period π 5. no y intercept 6. the vertical asymptotes occur at x..., π, 0, π, π,... nπ, 7. range: ( ) y cot x cot θ cot(θ + 80 k) or cot(θ + πk) where k is an integer. Problems: Worksheets B & C (all) of
15 Section Signs of Trig Functions sin Θ opp hyp adj cos Θ + hyp + opp + tan Θ + adj + sin Θ opp hyp adj cos Θ hyp + opp + tan Θ adj sin Θ opp hyp adj + cos Θ hyp + opp tan Θ + adj sin Θ opp hyp adj + + cos Θ + hyp + opp tan Θ adj + To summarize: Quadrant I Quadrant II Quadrant III Quadrant IV ALL Positive SINE Positive TANGENT Positive COSINE Positive A CAST diagram Or All Students Take Calculus 4 of
16 Section Signs of Trig Functions Example : If sinθ < 0 and cosθ > 0, in which quadrant must θ lie? Sin is negative in Quadrants III and IV Cos is positive in Quadrants I and IV Therefore, must be in Quadrant IV for both to be true Worksheet examples: #6 sec 540 sec 80 cos 80 #4 7 π π π cot cot 4 π + cot Could use graphs to find cos80 #0 #6 #48 Given: sin Θ, tan Θ cos Θ cot Θ cos ec Θ sec Θ 5 Given: sin Θ, θ in Quadrant III 5 + x ( ) x , x sin Θ cos Θ tan Θ cos ec Θ sec Θ cot Θ 5 5 cos Θ Given: sec Θ, tanθ > 0, so sin Θ cos Θ tan Θ cos ec Θ sec Θ cot Θ Problems: Worksheet A: all If cos is neg and tan is pos, must be in Quadrant III. ( ) +, y 4 (must be negative) y y 5 of
17 Section 4 Trig Equations We first find a principle value (PV) and then a secondary value (SV) in a different quadrant. Ex. : sin Θ / Sine is positive in Quadrants I and II PV 0 SV 50 There are infinite answers that satisfy sin Θ /, but there are just answers in the range 0 to 60. The general solutions are n and n. There are 4 solutions in range 0 70º: 0º, 50º, 90º, 50º Ex. : cos Θ / cos is negative in Quadrants II and III PV 0 SV n and n. Ex. : tan Θ tan is positive in Quadrants I and III PV 45 SV n and n. Which is the same as: n 6 of
18 Section 4 Trig Equations Ex. 4: cos Θ or cos Θ cos is positive in Quadrants I and IV PV 0 SV n and n. Ex. 5: sin Θ Sin 90 PV 90 SV none n Or using graph: Ex. 6: tan Θ 0. 4 NEED A CALCULATOR Θ tan ( 0. 4 ). 8 tan is negative in Quadrants II and IV PV θ SV θ n and n. Problems: Worksheet 4 (all) Review: Worksheet 5: all Note: there is no Section 5 7 of
19 Section 6. Sine Rule There are four general types of triangles we have to solve:. One side, two angles known. Two sides and included angle known SAS. Two sides and an opposite angle known 4. Three sides known Law of Sines: sin A a sin B sin C OR b c a sin A b sin B c sin C Usually, α, β, and γ are used for angles A, B, and C Law of Sines applies to AAS and ASA triangles And may of may not apply to SSA triangles And not to SAS or SSS triangles (at least directly) 8 of
20 Section 6. Sine Rule Ex : Solving an SAA (AAS) triangle: A7º, C4º, a6.4 B80º 4º 7º66º a b sin A sin B 6. 4 b sin 7 sin sin 66 b sin Ex : Solving an ASA triangle: a c sin A sin C 6. 4 c sin 7 sin sin 4 c 4. 6 sin 7 A4º, b4cm, C6º B c sin C c sin 6 4 c b sin B 4 sin 75 sin sin 75 Ex : Solving SSA ( ASS ) triangles: Ambiguous cases, can have one, two, or even zero solutions. ) if a < h < b, then no triangle ) if a h, then right triangle ) if a > h and a < b, then two possible triangles 4) if a 4 > h and a 4 > b, then one triangle possible a sin A a sin 4 4 a b sin B 4 sin 75 sin 4 sin Note: h sin α or h b sin α b Problems: Worksheet 6: 6 9 of
21 Section 6. Sine Rule Ambiguous Cases Usually, α, β, and γ are used for angles A, B, and C Ex : Solving an SSA triangle: b4, c, β40 As b > c, then only one triangle possible h sin 40, h sin sin C sin 40 sin 40, sin C not 5. because: > 80 (too big) C sin A a 4 sin. sin 40 Ex. #, a b, c, β40 (also SSA) h sin as >h but <, there are two possible triangles sin C sin 40 sin 40, sin C C sin or C if C 74.6, then A 80º 40º 74.6º 65.4 a, a. 8 sin sin 40 if C 05.4, then: A 80º 40º 05.4º 4.6 a, a. 77 sin 4. 6 sin 40 0 of
22 Section 6. Sine Rule Ambiguous Cases Ex. #4 a, b7, α70 (another SSA) h sin 70, h 7 sin As a < h, then no possible solutions If you try to solve this problem: The sine of an angle is never > or < sin 70 sin B 7 sin 70, sin B. 9 7 but sin B must be between and B sin. 9 no solution Problems: Worksheet 6: 7 8 of
23 Section 7. Law of Cosines a b c Law of Cosines: b a a + c + c + b bc cos A ac cos B ab cos C Ex : Solving an SAS triangle: a, c, β0 Cannot use Sine Rule, so use Cos Rule. ) b a + c ac cos B b + ( )( ) cos 0 b 5 b. 0 4 cos ) A ) Now we can use the Sine Rule: sin C sin 0 SEE NOTE BELOW. 0 sin 0 sin C C sin NOTE: Ensure when you find an angle using the sine rule that you find the smallest (acute) angle, as even if the angle is obtuse, the sine rule will always give you an angle in the first quadrant and so the answer could be wrong. The cosine rule will always give you the correct answer if you do it right! Ex : Solving an SSS triangle: a, b, c ) a b + c bc cos A + ( )( ) cos A cos A 4 cos A cos A / A cos ( / ) Problems: Worksheet 7: all ) per the Sine Rule: sin sin B, and B 70.5 ) C of
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