A mathematics PowerPoint by Eric Zhao

Trigonometry is the study and solution of Triangles. Solving a triangle means finding

the value of each of its sides and angles. The following terminology and tactics will be

important in the solving of triangles.

Pythagorean Theorem (a2+b2=c2). Only for right angle triangles

Sine (sin), Cosecant (csc or sin-1)

Cosine (cos), Secant (sec or cos-1)

Tangent (tan), Cotangent (cot or tan-1)

Right/Oblique triangle

A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent.

Let us look at this triangle…

ac

bө A

B

C

Given the assigned letters to the sides and angles, we can determine the following

trigonometric functions.

The Cosecant is the inversion of the sine, the secant is the inversion of

the cosine, the cotangent is the inversion of the tangent.

With this, we can find the sine of the value of angle A by dividing side a by side c. In order to find the angle itself, we must take the sine of the angle and invert it (in other words, find the cosecant of the sine of the angle).

Sinθ=

Cos θ=

Tan θ=

Side Opposite

Side Adjacent

Side AdjacentSide Opposite

Hypothenuse

Hypothenuse

=

=

= a

bca

b

c

Try finding the angles of the following triangle from the side lengths using the trigonometric ratios from the

previous slide.

610

8θ A

B

C

α

β

Click for the Answer…

The first step is to use the trigonometric functions on angle A.

Sin θ =6/10

Sin θ =0.6

Csc0.6~36.9

Angle A~36.9

Because all angles add up to 180, B=90-11.537=53.1

C

2

34º A

B

α

β

The measurements have changed. Find side BA and side AC

Sin34=2/BA

0.559=2/BA

0.559BA=2

BA=2/0.559

BA~3.578

The Pythagorean theorem when used in this triangle states that…

BC2+AC2=AB2

AC2=AB2-BC2

AC2=12.802-4=8.802

AC=8.8020.5~3

When solving oblique triangles, simply using trigonometric functions is not enough. You need…

The Law of Sines

C

c

B

b

A

a

sinsinsin

The Law of Cosines

a2=b2+c2-2bc cosA

b2=a2+c2-2ac cosB

c2=a2+b2-2ab cosC

It is useful to memorize these laws. They can be used to

solve any triangle if enough measurements are given.

a

c

bA

B

C

When solving a triangle, you must remember to choose the correct law to solve it with.

Whenever possible, the law of sines should be used. Remember that at least one angle measurement must be

given in order to use the law of sines.

The law of cosines in much more difficult and time consuming method than the law of sines and is harder to memorize. This law, however, is the only way to solve a

triangle in which all sides but no angles are given.

Only triangles with all sides, an angle and two sides, or a side and two angles given can be solved.

a=4

c=6

bA

B

C

28º

Solve this triangleClick for answers…

Because this triangle has an angle given, we can use the law of sines to solve it.

a/sin A = b/sin B = c/sin C and subsitute: 4/sin28º = b/sin B = 6/C. Because we know nothing about b/sin B, lets start with 4/sin28º and use it to solve 6/sin C.

Cross-multiply those ratios: 4*sin C = 6*sin 28, divide 4: sin C = (6*sin28)/4.

6*sin28=2.817. Divide that by four: 0.704. This means that sin C=0.704. Find the Csc of 0.704 º.

Csc0.704º =44.749. Angle C is about 44.749º. Angle B is about 180-44.749-28=17.251.

The last side is b. a/sinA = b/sinB, 4/sin28º = b/sin17.251º, 4*sin17.251=sin28*b, (4*sin17.251)/sin28=b. b~2.53.

a=2.4

c=5.2

b=3.5A

B

C

Solve this triangle:Hint: use the law of cosines

Start with the law of cosines because there are no angles given.

a2=b2+c2-2bc cosA. Substitute values. 2.42=3.52+5.22-2(3.5)(5.2) cosA,

5.76-12.25-27.04=-2(3.5)(5.2) cos A, 33.53=36.4cosA, 33.53/36.4=cos A, 0.921=cos A, A=67.07.

Now for B.

b2=a2+c2-2ac cosB, (3.5)2=(2.4)2+(5.2)2-2(2.4)(5.2) cosB, 12.25=5.76+27.04-24.96 cos B.

12.25=5.76+27.04-24.96 cos B, 12.25-5.76-27.04=-24.96 cos B. 20.54/24.96=cos B. 0.823=cos B.

B=34.61.

C=180-34.61-67.07=78.32.

Trigonometric identities are ratios and relationships between certain

trigonometric functions.

In the following few slides, you will learn about different

trigonometric identities that take place in each trigonometric

function.

What is the sine of 60º? 0.866. What is the cosine of 30º? 0.866. If you look at the name of cosine, you can actually see that it is the cofunction of the sine (co-sine). The cotangent is the cofunction of the tangent (co-tangent), and the cosecant is the cofunction of the secant (co-secant).

Sine60º=Cosine30º

Secant60º=Cosecant30º

tangent30º=cotangent60º

Sin θ=1/csc θ

Cos θ=1/sec θ

Tan θ=1/cot θ

Csc θ=1/sin θ

Sec θ=1/cos θ

Tan θ=1/cot θ

The following trigonometric identities are useful to remember.

(sin θ)2 + (cos θ)2=1

1+(tan θ)2=(sec θ)2

1+(cot θ)2=(csc θ)2

Degrees and pi radians are two methods of showing trigonometric info. To convert

between them, use the following equation.

2π radians = 360 degrees1π radians= 180 degrees

Convert 500 degrees into radians.2π radians = 360 degrees, 1 degree = 1π radians/180,

500 degrees = π radians/180 * 500500 degrees = 25π radians/9

Write out the each of the trigonometric functions (sin, cos, and tan) of the following degrees to the hundredth place.

(In degrees mode). Note: you do not have to do all of them

1. 45º

2. 38º

3. 22º

4. 18º

5. 95º

6. 63º

7. 90º

8. 152º

9. 112º

10. 58º

11. 345º

12. 221º

13. 47º

14. 442º

15. 123º

16. 53º

17. 41º

18. 22º

19. 75º

20. 34º

21. 53º

22. 92º

23. 153º

24. 1000º

Solve the following right triangles with the dimensions given

5c

22A

B

C

920

18A

B

C

A

ac

13

B

C

52 º

c

128 º A

B

C

Solve the following oblique triangles with the dimensions given

1222

14A

B

C

a25

b28 º

A

B

C

31 º

15c

2435 º

A

B

C

5c

8A

B

C

168 º

1. 45º

2. 38º

3. 22º

4. 18º

5. 95º

6. 63º

7. 90º

8. 152º

9. 112º

10. 58º

11. 345º

12. 221º

13. 47º

14. 442º

15. 123º

16. 53º

17. 41º

18. 22º

19. 75º

20. 34º

21. 53º

22. 92º

23. 153º

24. 1000º

Find each sine, cosecant, secant, and cotangent using different trigonometric identities to the hundredth place

(don’t just use a few identities, try all of them.).

Convert to radians

52º

34º

35º

46º

74º

36º

15º

37º

94º

53º

174º

156º

376º

324º

163º

532º

272º

631º

856º

428º

732º

994º

897º

1768º

2000º

Convert to degrees

3.2π rad

3.1π rad

1.3π rad

7.4π rad

6.7π rad

7.9 rad

5.4π rad

9.6π rad

3.14π rad

6.48π rad

8.23π rad

5.25π rad

72.45π rad

93.16π rad

25.73π rad

79.23π rad

52.652π rad

435.96π rad

14.995π rad

745.153π rad

Creator

Eric Zhao

Director

Eric Zhao

Producer

Eric Zhao

Author

Eric ZhaoMathPower Nine, chapter 6Basic Mathematics Second edition

By Haym Kruglak, John T. Moore, Ramon Mata-Toledo