TRIGONOMETRY.doc

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TRIGONOMETRY

Trigonometry – that branch of mathematics that deals with the study of triangles. Angle – the figure formed by two half-lines (called rays) from a common point or origin

(called vertex ).. Its size is the amount of rotation from one of the rays (or sides) to the other.

Kinds of Angles:

1. Acute angle – an angle which measures between 0o and 900.2. Right angle – an angle formed by 2 perpendicular rays. It measures exactly 90o.3. Obtuse angle – an angle which measures between 90o and 180o.4. Straight angle – an angle formed by two rays extending in opposite directions. It

measures exactly 180o.5. Reflex angle – an angle greater than 180o but less than 360o.6. Perigon – an angle measuring exactly 360o.

Complementary angles – angles whose sum is 90o.

Supplementary angles - angles whose sum is 180o.Explementary angles - angles whose sum is 360o.

Triangle – a polygon of three sides. One important property of a triangle is that the sum of twoof its sides is greater than the third side and their difference is less than the third side.

Example: In triangle ABC, side AB = 8 m and side BC = 20 m. One possible dimension of side CA isa. 13 b. 11 c. 9 d. 7

Solution: Since the difference of two sides of a triangle is less than the third side, then

BC – AB < CA20 – 8 < CA

12 < CA ⇒ CA > 12 ⇒ ∴CA = 13

Kinds of Triangles (according to size of angles):

1. Right triangle – a triangle with a right angle.2. Oblique triangle - a triangle without a right angle. This is further classified into

a. Acute triangle – a triangle all of whose angles are acute.b. Obtuse triangle – a triangle with one obtuse angle.

Kinds of Triangles (according to relation of sides):

1. Scalene triangle – a triangle with no equal sides.2. Isosceles triangle - a triangle with two sides equal.3. Equilateral triangle – a triangle with all 3 sides equal. It is also equiangular.

Similar triangles – triangles where the corresponding angles are congruent and thecorresponding sides are proportional. Given two similar triangles, the ratio of their areas is the square of the ratio of any two corresponding sides.

Let A1 & A2 be area of smaller & bigger triangle,respectively.

222

2

1

=

=

=

z

c

y

b

x

a

A

A

Congruent triangles – triangles that are identical.



The Pythagorean Theorem:

In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. A set of three positive integers that satisfy the Pythagorean principle such as 3, 4 and 5 and8, 15 and 17 constitute what is called a Pythagorean triple.

Units for Measurement of Angles:1. Degree – a measure of an angle which is equal to 1/360 of a revolution .2. Radian – the measure of the central angle subtended by an arc of a circle which is

equal in length to the radius of the circle. 1 revolution = 2π radians.3. Mil - a measure of an angle which is equal to 1/6400 of a revolution.4. Grad (or Gon) - a measure of an angle which is equal to 1/400 of a revolution.

To convert from degrees to radians and vice-versa, we use the relation 180o = π radians.

Pairs of Cofunctions: Pairs of Reciprocal Functions:1. sine & cosine 1. sine & cosecant

2. tangent & cotangent 2. cosine & secant3. secant & cosecant 3. tangent & cotangent

Solution Of Right Triangles:

A + B = 90o

222

cba =+

hypotenuse

sideopposite=θ sin

sideopposite

hypotenuse=θ csc

hypotenuse

sideadjacent =θ cos

sideadjacent

hypotenuse=θ sec

sideadjacent

sideopposite=θ tan

sideopposite

sideadjacent =θ cot

Special Right Triangles:

Solution of Oblique Triangles:



A + B + C = 180o

Law of Sines:

In any triangle, the sides are proportional to the sines of the opposite angles, i.e.:

C

c

B

b

A

a

sinsinsin==

Law of Cosines:

In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice their product times the cosine of their included angle, i.e.

Abccba cos2222

−+=

Baccab cos2222−+=

C abbac cos2222−+=

Fundamental Trigonometric Identities:

The Quotient Relations:θ

θ θ

cos

sintan =

θ

θ θ

sin

coscot =

The Pythagorean Relations:

θ θ

θ θ

θ θ

22

22

22

csc1cot

sectan1

1sincos

=+

=+

=+

Sum and Difference Formulas:

( )

( )

( )φ θ

φ θ φ θ

φ θ φ θ φ θ

φ θ φ θ φ θ

tantan1

tantantan

sincoscossinsin

sinsincoscoscos

±=±

±=±

=±

Double-Angle Formulas:

θ

θ θ

θ θ θ

θ θ θ θ θ

2

2222

tan1

tan22tan

cossin22sin

sin211cos2sincos2cos

−

=

=

−=−=−=

Half-Angle Formulas:

θ

θ

θ

θ

θ

θ θ

φ φ

θ θ

φ φ

θ θ

cos1

sin

sin

cos1

cos1

cos1

2

tan

2

2cos1sin

2

cos1

2sin

2

2cos1cos

2

cos1

2cos

2

2

+

=−

=

+

−±=

−=

−±=

+=

+±=

or

or



Functions of θ − :

( )

( )

( ) θ θ

θ θ

θ θ

tantan

sinsin

coscos

−=−

−=−

=−

Functions of Complementary Angles:

( )

( )

( )θ θ

θ θ

θ θ

−°=

−°=

−°=

90cottan

90cossin

90sincos ( )

( )

( )θ θ

θ θ

θ θ

−°=

−°=

−°=

90seccsc

90cscsec

90tancot

Review Exercises in TRIGONOMETRY

1. Find the supplement of an angle whose compliment is 62o.

a. 28o b. 118o c. 152o d. 162o

2. To find the angles of a triangle, given only the lengths of the sides, one would use

a. law of tangents b. law of sines

c. orthogonal functions d. law of cosines

3. Which is true regarding the signs of the natural functions for angles between 90o and

180o?

a. cosine is negative b. sine is negative

c. secant is positive d. tangent is positive

4. What is the value of θ that will satisfy the equation sin2 θ + 4 sin θ + 3 = 0 if

0 ≤ θ < 2π?

a. π b. π/4 c. 3π/2 d. π/2

5. Given a triangle with angle C = 28.7o, side a = 132 units and side b = 224 units. Solve

for side c.

a. 110 units b. 125.4 units c. 95 units d. 90 units

6. One radian is equivalent to

a. 52o b. 45o c. 90o d. 57.3o

7. Given a right triangle where the hypotenuse is r, an acute angle θ, the side opposite θ

is “y” and the adjacent side “x”. Which of the following relations does not apply?

a. x2 + y2 = r 2 b. cos 2θ = 1 + 2 sin2 θ

c. sin 2θ = 2 sin θ cos θ d. 1 + tan2 θ = sec2 θ

8. What do you call an angle whose terminal side coincides with either the x- or the y-

axis?

a. reflex angle b. quadrantal angle c. co-terminal angle d. right angle

9. For an object which is below the eye of an observer, the angle which the line of sight



to the object makes with the horizontal is called

a. angle of depression b. angle of elevation

c. bearing d. vertical angle

10. Two sides of a triangle measure 6 cm and 8 cm and their included angle is 40o. Find

the third side.a. 5.144 cm b. 5.263 cm c. 4.256 cm d. 5.645 cm

11. The sum of the two interior angles of a triangle is equal to the third angle and the

difference of the two angles is equal to 2/3 of the third angle. Find the third angle.

a. 15o b. 75o c. 90o d. 120o

12. Two sides of a triangle are 50 m and 60 m long. The angle included between these

sides is 30o. What is the interior angle opposite the longest side?

a. 92.74o b. 93.74o c. 94.74o d. 91.74o

13. The measure of 1.5 revolutions counterclockwise isa. 540o b. 520o c. 90o d. – 90o

14. The product cot θ cos θ is also equivalent to

a. cos θ b. csc θ – sin θ c. sec θ – cos θ d. sin θ – csc θ

15. If sin A = 3/7, where A is an acute angle, what is the exact value of tan A?

a. 3/102 b. 20/103 c. 102/7 d. 20/107

16. Simplify:1csc

csc

1csc

csc

+

+

− θ

θ

θ

θ

a. 2 cot2 θ b. 2 tan2 θ c. 2 csc2 θ d. 2 sec2 θ

17. How many different values of x from 0o to 180o satisfies the equation

(2 sin x – 1)(cos x + 1) = 0?

a. 3 b. 0 c. 1 d. 2

18. An obtuse angle is ______ than a right angle.

a. lesser b. equal c. greater d. oblique

19. Which of the following is equivalent to csc θ cos3 θ tan θ?

a. cos θ b. 1 – sin2

θ c. sin2

θ d. 1

20. What is the value of the angle whose tangent is equal to 1 and the sine of which is a

negative quantity?

a. 45o b. 90o c. 225o d. 315o

21. An angle more than π radians but not less than 2π radians.

a. straight angle b. obtuse angle c. reflex angle d. vertical angle

22. Solve: cos2 θ + 2 sin θ + 1 = 0 given that 180o < θ < 270o.

a. 227.058o b. 219.047o c. –132.942o d. 255.036o



23. In general, for a triangle ABC, the equation (sin A)/a = (sin B)/b = (sin C)/c

represents

a. the law of cosines b. the law of sines

c. the law of tangents d. the reciprocal of the law of sines

24. The difference of the squares of the secant and the tangent of an angle is equal toa. 2 b. 1 c. 0 d. 3

25. In the fourth quadrant,

a. sin B > 0 and cos B < 0 b. sin B > 0 and cos B > 0

c. sin B < 0 and cos B < 0 d. sin B < 0 and cos B > 0

26. Sin 2A cos B – cos 2A sin B is equivalent to

a. sin 2(A – B) b. cos 2(A – B) c. sin (2A – B) d. cos (2A – B)

27. The cosecant function is negative in quadrants

a. III and IV b. I and II c. I and III d. II and IV

28. Sin (270o + β) is equal to

a. – cos β b. sin β c. – sin β d. cos β

29. If 0 < Arccot x <180o, then Arccot (-1/ 3 ) is

a. 30o b. – 60o c. 120o d. 150o

30. It is the measure of an angle equivalent to 1/400 of a revolution.

a. mil b. degree c. radian d. grad

31. It is the measure of an angle equivalent to 1/6400 of a revolution.

a. mil b. degree c. radian d. grad

32. 3200 mils is equal to how many degrees?

a. 45o b. 90o c. 180o d. 270o

33. Express 45o in mils.

a. 80 mils b. 800 mils c. 8000 mils d. 80,000 mils

34. Evaluate: sec(Arccot 2)

a. 5 b. 2/5 c. 2 d. ½

35. A triangle having three sides of unequal lengths is known as ______ triangle.

a. right b. oblique c. scalene d. equilateral

36. The measure of 2.25 revolutions clockwise is

a. – 835o b. 805o c. – 810o d. 810o

37. An angle greater than a straight angle and less than two straight angles is called

a. right angle b. obtuse angle c. acute angle d. reflex angle

38. Which of the following statements is correct?a. All equilateral triangles are similar. b. All right triangles are similar.



c. All isosceles triangles are similar. d. All rectangles are similar.

39. Explementary angles are angles whose sum is

a. 90o b. 180o c. 270o d. 360o

40. Solve for θ: θ = arccos (sin 2θ)a. 30o b. 45o c. 60o d. 15o

41. Solve for x: Arctan x + Arctan x

3

2=

4

π

a. 1 b. ½ c. 2 d. 3/2

42. An observer found the angle of elevation of the top of a tree to be 27o. After moving

10 m closer (on the same vertical and horizontal plane as the tree), the angle of

elevation becomes 54o. How high is the tree?

a. 8.65 m b. 7.53 m c. 7.02 m d. 8.09 m

43. If x and y are complementary angles, find the value of P if P = cos (540o – x) sin (540o – y) + cos (90o – x) sin (90o – y)

a. – sin 2x b. cos 2y c. cos 2x d. – cos 2y

44. A ladder 42 ft long is placed so that it will reach a window 30 ft high on one side of a

street; if it is turned over, its foot being held in position, it will reach a window 25 ft

high on the other side of the street. How wide is the street from building to building?

a. 63.1 ft b. 65.4 ft c. 71.8 ft d. 82.5 ft

45. Each leg of an isosceles right triangle whose perimeter is 30 inches is _____ inches.

a. 8.45 in. b. 8.79 in. c. 9.03 in. d. 9.72 in.

46. A person on a ship sailing due south at the rate of 15 miles an hour observes a

lighthouse due west at 3 p.m. At 5 p.m. the lighthouse is 52o west of north. How far

from the lighthouse was the ship at 4 p.m.?

a. 38.4 mi. b. 48.73 mi. c. 41.22 mi. d. 45.75 mi.

47. Which of the following is true?

a. cot (-θ) = cot θ b. cos (-θ) = - cos θ c. sec (-θ) = sec θ d. csc (-θ) = csc θ

48. From the top of a tower 63.2 ft high, the angles of depression of two objects situated

in the same horizontal line with the base of the tower, and on the same side of the

tower, are 31o16’ and 46o28’ respectively. Find the distance between the two objects.

a. 104.08 ft b. 60.04 ft c. 52.06 ft d. 44.04 ft

49. Cos (A + B) cos A + sin (A + B) sin A is equal to

a. sin B b. sin (2A + B) c. cos B d. cos (2A + B)

50. A vertical pole 35 ft high, standing on sloping ground, is braced by a wire which



extends from the top of the pole to a point on the ground 25 ft from the foot of the

pole. If the pole subtends an angle of 30o at the point where the wire reaches the

ground, how long is the wire?

a. 54.3 ft b. 56.7 ft c. 50.8 ft d. 59.5 ft

51. The sides of a triangular lot are 130 m, 180 m and 190 m. What is the area of the lot?a. 11,260 m2 b. 11,225 m2 c. 10,250 m2 d. 14,586 m2

ANSWER KEY:

1. c 11. c 21. c 31. a 41. b 51. b

2. d 12. b 22. a 32. c 42. d

3. a 13. a 23. b 33. b 43. b

4. c 14. b 24. b 34. b 44. a

5. b 15. b 25. d 35. c 45. b

6. d 16. d 26. c 36. c 46. c

7. b 17. a 27. a 37. d 47. c

8. b 18. c 28. a 38. a 48. d

9. a 19. b 29. c 39. d 49. c

10. a 20. c 30. d 40. a 50. a

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