Right Triangles and Trigonometry

By Mathew Chacko and Jessica Wu

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Overview of Concepts

● Sine, Cosine, and Tangent help provide ratios that find the interior angles and side lengths of triangles.

● SOHCAHTOA use when right triangle● first, you must use an angle that is not right (90)● determine whether to use sin, cos, or tan based

on the side lengths you know (opp and hyp=sin, adj and hyp=cos, adj and opp=tan)

● put sin,cos, or tan in front of said angle measure and set equal to trig ratio

● find a proportion using unknown angle and trig ratios (sin, cos, tan)

● tan(A)=opp/adj, sin(A)=opp/hyp, cos(A)=adj/hyp● if using sine, find the opp and hyp of said angle, same for

cosine and tangent● after finding a fraction using given numbers, apply inverse to

the sign● example below (easy)

Inverse SOHCAHTOA

A

16

20

find measure of <Ause tan b/c you know opp and adj side compared to <Atan(A)=16/20=4/5 apply inversetan-1(4/5) is about 38.7<A≈38.7 degrees

Law of Sine● A,B,C are angle measurements● a,b,c are side lengths● law of sine is sinA/a= sinB/b= sinC/c● first proportion you use must have given a side and angle● proportion you are solving for must have only one known

side or angle● angle measure over side set equal to angle/side (unknown)● then cross multiply to isolate unknown variable then solve● remember it only finds acute angles

Law of Cosine

● law of cosine is a²=b²+c²-2bc cos(A), b²=a²+c²-2ac cos (B), c²=a²+b²-2ab cos (C)

● one side squared set equal to other two sides squared minus two times those sides multiplied together than cosine of angle corresponding with original side

● only need to use if only know all the sides or SAS● example below (medium)

A

B

C60°

10

a8

FIND THE MISSING SIDE a²=64+100-2(80)cos(60)164-80=84a²=2√21 ≈9.2a=9.2now that you know all the sides and an angleyou can solve the the rest of the angles using law of sines

Two Triangles, One, or None● must test if a SSA (butt) triangle● example, if you use sine and find an angle to be 30 degrees, you must

also test the theory using 180 minus 30 which equals 150 b/c sine only finds acute angles

● by using side lengths, determine if one neither or both are plausible● if a>b then A>B, if both answers work, then two triangles, if only one

than one triangle● no solution if sine ends up greater than one● example below (difficult)

c a=10

b=12

A=24°

B

C

FIND THE OTHER TWO ANGLESSSA triangle so solve for more than one possibilitysin24/10=sinB/12=sinB= sin24/10<B≈29 degrees or 151 degreesbased on triangle, 12>10 so it could be either so two triangles180-24-29=127, 180-151-24=6final angle= 6 or 127 degreesA=24 degrees B=29 or 151 degrees C=6 or 127 degrees

Connections to Other Units

● trigonometry relates to proportions b/c students make proportions with sine, cosine and tangent

● example would be sin 62/32=sin76/b sin62(b)=sin76(32) b=33.5

● also people use SOHCAHTOA when you don’t know two sides of a right triangle

● if two sides are given, then just apply pythagorean theorem● which connects to unit 7a● you may sometimes

need to use trigonometry to find the area/volume of a triangle in a 3-D or 2-D shape

Common Mistakes

● many students forget the ratios for sine, cosine, and tangent a simple and easy trick to help memorize these confusing ratios would be the word “SOHCAHTOA” pic on the right

● students also forget to solve for more than one solution when you have a “butt” triangle or side side angle triangle (SSA)

● remember in previous units when we prove triangles congruent we never use SSA so you must solve for all possible solutions if SSA

Real Life Scenario You are building a ramp for a moving company that slants upward at a certain angle degree. You know that the side on the ground is 10m and the slant upwards is 12m. You also know that the angle on the ground is 24°. Find the rest of the dimensions.

A

B

C

c a=12m

b=10m

24°

Work:(use law of sine)1. sin24/12= sinB/10(multiply the proportionality)2.sin24x10= sinBx12(use calculator to reduce equation)3.angle B is about 19.8 degreees(not 180-19.8 b/c a>b)4.180 - two <’s= 136.2 degrees5.sin 24/12=sin 136.2/c(cross multiply)6.c is about 20.4

a=12 b=10 c=20.4<A=24 <B=19.8 <C= 136.2

GOOD LUCK ON YOUR FINAL