Find the trigonometric functions of the given angles

Question

1. Using the sides of the triangles, find the trigonometric functions of the given angles indicated in the table below.

____________________________________________________________________________

| |Angle A in | Angle D in | Angle G in | Angle C in | Angele D in | Angle E in | Angle B in |

|______|_∆ABC__ |__∆GED__|___∆GEI__|__∆EDC___|__∆ABD___|__∆EFC___|__∆BED___|

|sin θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

|cos θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

|tan θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

|csc θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

|sec θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

|cot θ | | | | | | | |

|______|________|_________|__________|_________|__________|__________|__________|

Here is the link for the image of the triangle and a not so deformed table:

https://miro.com/welcomeonboard/eTdCVE1jYUFBZFVIS3BaNWJ0UEI1OUVaTGlDOGoxTTBKOG12aHBqZXc3WFpoVFhxTzBQeGtpT2JLbmRSdVZKS3wzNDU4NzY0NTE5NDEwNzg2Mjg0?invite_link_id=157293958503

Answer 1

Tricky problem!

I believe what this problem is asking is for you to find the cos, sin, tan, etc of certain angles in the picture. Since we're only given side lengths, we have to figure out what the cos, sin, tan of each angle is by its corresponding ratio, such as sin(theta)=opposite/hypotenuse etc. For example, under the sin(theta) box for Angle A, we would put the ratio of BD to AD.

The problem doesn't give you enough values of the side lengths to automatically go ahead and start finding angles. We have to figure out some more side length measurements first. So lets look at what we can figure out visually and using the Pythagorean theorem:

1. DC must equal EF since EFCD is a triangle. So EF=12
2. ED must equal FC since EFCD is a triangle. So ED=10.
3. Now since we know the sides DC and ED, we can find the third leg of the triangle EDC, which is the hypotenuse EC. We use the Pythagorean theorem to find that EC²=ED²+DC² or EC²=10²+12². This gives us that EC is the square root of 244, or 2 times the square root of 61.
4. We can do the same for the triangle GEI. We know the two legs, GE and IE, are 5 and 3, so we can find the hypotenuse GD using the Pythagorean theorem. This gives us GD²=3²+5² so GD is the square root of 17.
5. We're told that AE = EC. This tells us that the side AC is double the length of EC, so AC is 4 times the square root of 61.
6. We can use the Pythagorean theorem again to find BC, since we know AB and AC. AB²+BC²=AC². This gives us BC²=976-400=576. BC is 24.
7. Now that we know BC, we can find BD by removing the DC portion of the line. BD=BC-DC=24-12 so BD=12.
8. We can use the Pythagorean theorem again to find AD. AD²=AB²+BD²=400+144. AD is the square root of 544.
9. Now that we know BD and ED we can find BE from the Pythagorean theorem. BE²=BD²+DE²=144+100. BE is the square root of 244.
10. The last side we have to find is GD. We know ED and GE, so we can find GD using the Pythagorean theorem. I assume by now you can calculate this yourself.

Now we've found all the sides we needed! Let's start filling in the table. Each column corresponds to one angle in a certain triangle. We have to find the sin, cos, tan, etc. of this angle using the ratio of the sides of the triangle. Lets start with the first column, angle A in ABC. We'll be finding sin(A), cos(A), etc.

1. sin(A). Sin is opposite over hypotenuse. From the perspective of angle A, the opposite side is BC. The hypotenuse, which is opposite the right angle, never changes regardless of what angle's perspective we are using. In this triangle, the hypotenuse is AC. Therefore sin(A)=BC/AC or 24/(4*sqrt(61))=.77.
2. cos(A). Cos is adjacent over hypotenuse. To angle A, adjacent is AB. Cos(A)=AB/AC=20/(4*sqrt(61))=.64.
3. tan(A). Tangent is opposite over adjacent, or sin over cos. Since we know sin(A) and cos(A) already, its easiest to find tan(A) from these values. Tan(A)=.77/.64=1.2
4. csc(A) is hypotenuse over opposite, or 1 divided by sin. We know sin(A) already so csc(A)=1/sin(A)=1/.77=1.30.
5. sec(A) is hypotenuse over adjacent or 1 divided by cos. We know cos(A) so sec(A)=1/cos(A)=1/.64=1.56.
6. cot(A) is adjacent over opposite. This is also csc/sec or soc/sin or 1/tan. Cot(A)=1/tan(A)=1/1.2=.833.

And there you have it! We have the needed side values and how to plug them into the table. The process is the same for the rest of the columns, pay attention to which side is opposite and which side is adjacent for each angle.