Solving Varying Angle Problems Using Sine and Cosine Laws

let's go to number five this problem is about varying angles the following diagram shows a three tons

a b and a c tongue b is a nine kilometer from tongue a which means this is a nine

a bearing of 4 75 or degree which means this one is a 75 degree tongsey is a 13 kilometer from tomb

on a bearing of 107 degree so we have here's 13 kilometers

and this angle is a 107 degree

find angle a b c this angle is a we are looking for think about

say letter c you see here this is the letter c

so think about letter c now you see this is the letter c therefore

let's label here is a theta for this angle theta

we will use 180 minus 75 degree

because they are consecutive interior angles together 105 degree

then angle abc we will use a 360 degree minus 107 degree minus 105 degree

so you will get the 140 degree let's go to 5b

find the distance from tongue a to tongue c first of all we need to connect a nsc

then we need to label this is a 148 degree we could see for this

148 degree is a included angle for included angle you got to use the cosine log

for the cosine law let's write down c squared equals a squared plus b squared minus 2 a b

cosine c therefore for

ac squared equals the corresponding angle is a 1 0

degree therefore you will get

9 squared plus 13 squared minus 2 times 9 times 13 cosine 148

degree you will get the ac equals 21.2

kilometers now let's go to five or six use the sine rule to find the angle a

say b a say b is this angle we know this is a

21.2 kilometers therefore for sine rule we know is the a over sine a

equals b over sine b corresponding sides over corresponding angle

let's define the variable first let angle a say b

equals alpha the corresponding side of for this angle

alpha is 9. so we will get the 9 over sine alpha equals

we have 21.2 over sine 148

degree then cross multiply the same line with the variable go to

denominator so we have 21.2 9 times the sine

148 degree so you can write down alpha equals

inverse sine of 9 times the

sine 148 degree over

21.2 equals 13.0 degree

let's go to 6a a shape is a ceiling north from a point a

towards point d point c is 1 78 kilometer north of

a pond d is a 80 kilometer north of

sea there is an island at e the bearing of yi from a

is a 58 degree the bearing of yi from say is

124 degree this is shown in the following diagram find the bearing of

a from e according to the information we know

this is 178 this is a 80 and uh this angle

50 58 degree this angle

is 124 degree we are looking for

the bearing of a from e so we will draw the

north line here this is north line then

this angle we are looking for let's label as a theta this angle is a

exterior angle of this triangle a

e c therefore let's label this angle

is a alpha since this angle is the exterior angle of our

triangle therefore 124 equals 58 plus alpha

in other words alpha equals 124 degree minus 58 degree and this angle

let's label as a beta for this beta

you could say this is a c so we have a

consecutive angles then we know 124 plus beta will be

180 because they are consecutive interior angles so for the 6a

we can get the beta first is 180 degree minus 124 degree that is a

56 degree for alpha equals 124 degree minus

58 degree you will get 66 degree

so for theta equals 1 so for theta equals 360 degree minus

56 degree minus 66 degree equals 238 degree

so the varying angles of a

from e is 238 degree

let's go to six b finder say e c

is here we already know the alpha

a 66 degree since a 66 degree and a 58 degree they

are not interior angles so we have to use

since for this alpha and a 58 they are not

included angles we got to use the sun law

for the sun law we know that is a over sine a equals b over sine b

make sure it's a corresponding side over corresponding angle

therefore you will have 178 over sun

66 a degree equals say yi over sun

58 degree cross multiply say e

equals same line with the variable go to denominator so sun

66 degree 178 degrees sun 58 degree

equals so you get 165 kilometers

for c e

let's label here 165 kilometers for six say

find d e first of all we need to connect d e and we already know this angle

124 degree and for the seiyee is 165 kilometers

so 1 24 degree is a included angle

for included angle you need to use the cosine law for the cosine law that is a c squared

equals a squared plus b squared minus 2 a b cosine c

therefore for 6 is a we're going to use the d e squared equals

80 squared plus 165 squared minus 2 times 80

times 165 or cosine

124 degree for d e equals so we have

80 squared plus

165 squared minus 2

times 80 times 165

times cosine 124 degree

enter then we will do the square root second x

square drop this down here your answer will be 220 kilometers so de

equals uh 220 kilometers [Music] you