MCR3UP - Unit 4 - Lesson 3
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St Edmund Campion Secondary School**We aren't endorsed by this school
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MATH MCR3UP
Subject
Mathematics
Date
Dec 16, 2024
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13
Uploaded by Nefth
For any point P(x, ¥) in the Cartesian plane, the trigonometric ratios for angles in standard position can be expressed in terms of x,y,and r. singd = Y
For any point P(x, y) in the Cartesian plane that intersects the unit circle, the trigonometric ratios for angles can be expressed in terms of x,y, and r. sinf =3 = Y tan & :l X 270° COS@' (S X“Cbo\fo(t/tcula 57C :[«+ Smfi L\f \r~Coor‘c/1L,\q/Q/{
We know the x-coordinate of where the terminal arm Ll/u{- chl & Intersects the unit circle is equivalent to the cosine ratio and the <{<ZU‘7~ Lfs".\,- -+ (K, Y ) y-coordinate is equivalent to the sine ratio. y b L 435 —l&‘% F\ e / XS a p PR RN Notice that both 457 and 135° have the same QE fi) | vati>S . Since the angles fall in quadrants(_ and D | & Z—respectively, they will have the exact same y-coordinate but 1 == Ll Cae 1 the x-coordinates will have the same absolute value but will be opposite signs. . D sin135°= S lds" o . Cos 135°% = — Css 48 2 1 The important takeaway from this is that there are QL_‘ %\ eS between 0° and 360° that have the gxact same ratio. Using reference angles and the CAST rule, we'tan make sure to always find both possible angles between 0° and 360° that have the same trigonometric ratio.
Example 1: Solve the following equations for 0° < 8 < 360°. Round answers to the nearest tenth of a degree. Do Sm_'go._-n a) sins P e H@”%flir{ b)tan§ = 2.1 < A b=sw(-07)S ? b= et 2.0 = —44.3° = (s> —_— S 3 @ 360" —H4.3 ’ % -] o = 3157 - IRobL%S o > 3 +¢v,3 - 24457 =22¢4.3% —
Example 2: The point (—7,19) lies on the terminal arm. Find the angle to the terminal arm (principal angle, 8) and find the related acute angle (reference angle, f). T— FonfB = g:fi < A =m<:2) = “j ° _ § =196 -(q.8° , = 118,2° muo,2=(—'7i‘)
Example 3: The point P(5, 11) lies on the terminal arm of angle & in standard position. Draw a sketch of angle 8, determine the exact value of r, determine the primary trig ratios for angle 8, then calculate 8 to the ‘—% nearest tenth of a degree. V‘L _ E'L " |7_ Sastel ) . Y o_o = 1Ye Sln e’ = — == - o o= Jivh -X _ 5 Cos 9 -Z = - [C8
Example 4: Solve the following equation for G%fi_fl". I} 3cos6+1=0 Find refvua anle C¢>56'= —_‘_[_ B—‘ Qas" i = e :75‘&} . > _ Q = 25068 8 /t)f{S ) 150+ S D :)505 T C
Determine another angle beowesn 07 and 360° that has the same tangent ratio as tan(14°). Determine another angle between 07 and 360° that has the same sine ratio as sin(35°). Determine another angle beween 07 and 3607 that has the same cosine ratio as cos(1477). Determine another angle between 0% and 360° that has the same cosine ratio as cos(254°).
The paint P(Z, —fi) is an the terminal arm of the angle @ in standard position. Nhat are the exact values of the primary trigonometric ratios for 67 b. Calculate the measure of 8, to the nearest degree. c. Determine the measure of all coterminal angles to #in the interval —360° < 8 < T20". s Given P(2,~6), 7 = 2,y = —6, and b. Since the cosine ratio is positive, we can use it to find the related acute angle a. 1 2?4y cos{a) = Vo lifi = cos l( L ) - m VIO — 2/10 ars 72° The primary trigonametric ratios are Since P(2, —6) isin Quadrant IV, # = 360° — a = 288", . ] 6 3 . () . S sin(f) T 2410 V10 2 1 . === = d cos(f) Vi Vi an,
Question: The angle 960° is in standard position. In which guadrant, or on which axis, does the terminal arm lie? Question: The angle —150° is in standard position. In which quadrant, or on which axis, doeg the terminal arm lie? Question: The angle —450” is in standard position. In which guadrant, or on which axis, does the terminal arm lie? Question: The angle 1320° is in standard position. In which quadrant, or on which axis, does the terminal arm lie?
c. We can add integer multiples of 360° to find coterminal angles to . In the interval —360° < @ < T20°, the coterminal angles are approximately 288° — 360" = —72" and 2887 4 360" = 648"
The point P (—4, —5) is on the terminal arm of the angle #in standard position. Determine all possible values for @where —360° < 8 < 360°. Round your answers to the nearest degres. Since £ = —4 and y = —5, we know that the terminal arm of #is in Quadrant IIL P-4, -5) Let & be the related acute angle to 6 and we can write the positive tangent ratio equation. 5 tan(a) - 5 P(—4, -5) 5 =tan (= wovn () 72 517 The positive value of @in Quadrant M is § = 180° 4 @ == 231°. The negative value of 8in Quadrant III is coterminal, so f == 231° — 360° = —129°.