PLANE AND SPHERICAL TRIGONOMETRY 5.

A certain angle has an explement 5 times the

supplement. Find the angle.
I. Plane Angle and Angle Measurement a. 67.5 deg c. 135 deg
b. 108 deg d. 58.5 deg
A plane angle is determined by rotating a ray about its 6. What is the reference angle and one coterminal
endpoint called vertex. angle, respectively of 135°?
a. -45°, -225° c. 45°, 225°
b. -45°, 225° d. 45°, -225°

II. Right Triangle

where: a = opposite side

b = adjacent side
Angle Measurement c = hypotenuse
Null Angle 0°
Acute Angle 0°< θ < 90°
Right Angle 90°
Obtuse Angle 90°< θ < 180° The Pythagorean Theorem
Straight Angle 180°
Reflex Angle 180°< θ < 360° “In a right triangle, the square of the length of the
Perigon (Full revolution) 360° hypotenuse is equal to the sum of the squares of the
lengths of the legs” (c2 = a2 + b2)
1 revolution = 360 degrees
= 2π radians Note:
= 400 gradians In any triangle, the sum of any two sides must be
= 6400 mils greater than the third side; otherwise no triangle can be
formed.
QUESTIONS: If, c2 = a2 + b2 The triangle is right
1. The measure of 2.25 revolutions c2 > a2 + b2 The triangle is obtuse
counterclockwise is c2 < a2 + b2 The triangle is acute
a. -835° c. -810°
b. 805° d. 810° The Six Trigonometric Functions
2. 4800 mils is equivalent to ____ degrees.
a. 135 c. 235 Using the length of the sides of a right triangle, six ratios
b. 270 d. 142 are formed that define the six trigonometric functions.
3. An angular unit equivalent to 1/400 of the
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎
circumference of a circle is called sin θ = = cot θ = =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜
a. degree c. radian 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ
b. mil d. grad cos θ = = sec θ = =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ
tan θ = = csc θ = =
Angle Pairs 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜

∠ A + ∠ B = 90° Complementary angles Angle of Depression and Elevation

∠ A + ∠ B = 180° Supplementary angles
∠ A + ∠ B = 360° Explementary angles

Reference Angle

A reference angle θ for an angle α in standard position is

the positive acute angle between the x-axis and the
terminal side of angle α.

Coterminal Angle
The angle of elevation of the top of the tower from a point
Two angles which when placed in standard position, have 40 m from its base is the complement of the angle of
coincident terminal sides are called coterminal angles. elevation of the same tower at a point

𝜃 = 𝛽 + 𝑘(360°) QUESTIONS:
where: θ = the coterminal angle of angle β 7. The sides of a triangular lot are 130 m, 180 m and
k= an integer 190 m. This lot is to be divided by a line bisecting
the longest side and drawn from the opposite
QUESTIONS: vertex. Find the length of this line.
4. Find the complement of the angle whose a. 120 m c. 122 m
supplement is 152°. b. 130 m d. 125 m
a. 28° c. 118° 8. 120 m from it. What is the height of the tower?
b. 62° d. 38° a. 59.7 c. 69.3
 b. 28.5 d. 47.6 from B the bearing of C is 26 degrees N of E.
9. One leg of a right triangle is 20 cm and the Approximate the shortest distance of tower C to
hypotenuse is 10 cm longer than the other leg. the highway.
Find the length of the hypotenuse. a. 364 m c. 394 m
a. 10 c. 25 b. 374 m d. 384 m
b. 15 d. 20 13. A PLDT tower and monument stand on a level
10. A man finds the angle of elevation of the top of a plane. The angles of depression of the top and
tower to be 30 degrees. He walks 85 m nearer the bottom of the monument viewed from the top of
tower and finds its angle of elevation to be 60 the PLDT tower are 13° and 35° respectively. The
degrees. What is the height of the tower? height of the tower is 50 m. Find the height of the
a. 76.31 m c. 73.16 m monument.
b. 73.31 m d. 73.61 m a. 33.51 m c. 47.30 m
b. 7.58 m d. 30.57 m

III. Oblique Triangle IV. Area of Triangle

An oblique triangle is a triangle that contains no right Given base & altitude:
angle.
1
𝐴= 𝑏ℎ
2

Given all sides:

𝐴 = √𝑠 (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)

Centroid – intersection of all the medians of a triangle Equilateral triangle:

Angle bisector – bisects an angle
Median – vertex to midpoint of opposite side 𝐴=
√3
𝑎2
Orthocenter – intersection of all altitude of a triangle 4
Incenter – intersection of all the angle bisectors of a
triangle
Given 2 sides & included angle:
Sine Law 1
𝐴= 𝑎𝑏 sin 𝜃
2
𝑎 𝑏 𝑐
= =
sin 𝐴 sin 𝐵 sin 𝐶
Given 2 angles & included side:
Use Sine Law if:
 Given two angles and any side 𝐴=
𝑥 2 𝑠𝑖𝑛𝜃 sin 𝛼
 Given two sides and an angle opposite one of 2 sin 𝛽
them

Cosine Laws Circle inscribed in a circle:

Standard Form: Alternative Form:

𝐴 = 𝑟𝑠
2 2 2 𝑏 2 + 𝑐 2 −𝑎2
𝑎 = 𝑏 + 𝑐 − 2𝑏𝑐 cos 𝐴 cos 𝐴 =
2𝑏𝑐
2 2 2 𝑎2 + 𝑐 2 −𝑏 2 Circle circumscribing a triangle:
𝑏 = 𝑎 + 𝑐 − 2𝑎𝑐 cos 𝐵 cos 𝐵 =
2𝑎𝑐
𝑎2 + 𝑏 2 −𝑐 2
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶 cos 𝐶 =
2𝑎𝑏
𝑎𝑏𝑐
𝐴=
Use Cosine Laws if: 4𝑟
 Given three sides
 Given two sides and their included angle
Triangle with escribed circle:
QUESTIONS:
11. In a triangle, find the side c if the angle C = 100°, 𝐴 = 𝑟 (𝑠 − 𝑎)
side b = 20 and side a = 15.
a. 28 c. 29
b. 27 d. 26
12. Points A and B 1000 m apart are plotted on a
straight highway running east and west. From A,
the bearing of a tower C is 32 degrees W of N and
 QUESTIONS: Inverse tangent function
14. Given a right triangle ABC, C is the right angle, 𝑦 = 𝑎𝑟𝑐 tan 𝑥 iff tan y = x
BC = 4 and the altitude to the hypotenuse is 1
unit. Find the area of the triangle. QUESTIONS:
a. 2.0654 sq. u. c. 1.0654 sq. u. 19. If sec 2A = 1/sin 13A, determine the angle A in
b. 3.0654 sq. u. d. 4.0654 sq. u. degrees.
15. In a given triangle ABC, the angle C is 34°C, a is a. 5° c. 3°
29 cm and side b is 40 cm. Solve for the area of b. 6° d. 7°
the triangle. 20. Simplify the expression 4 cos y sin y (1-2 sin2 y)
a. 324.332 cm2 c. 317.15 cm 2 a. sec 2y c. tan 4y
b. 344.146 cm 2 d. 343.44 cm2 b. cos 2y d. sin 4y
16. A right triangle is inscribed in a circle such that 21. If sin A = 2.511x, cos A = 3.06x and sin 2A =
one side of the triangle is the diameter of a circle. 3.939x, find the value of x?
If one of the acute angles of the triangle a. 0.265 c. 0.562
measures 60 degrees and the side opposite that b. 0.256 d. 0.625
angle has length 15, what is the area of the circle? 22. Solve for x if tan 3x = 5 tan x.
a. 175.15 c. 235.62 a. 20.705° c. 15.705°
b. 223.73 d. 228.61 b. 30.705° d. 35.705°
17. The sides of a triangle are 8 cm, 10 cm, and 14 23. If arc tan 2x + arc tan 3x = 45 degrees, what is
cm. Determine the radius of the inscribed and the value of x?
circumscribing circle. a. 1/6 c. 1/5
a. 3.45, 7.14 c. 2.45, 8.14 b. 1/3 d. ¼
b. 2.45, 7.14 d. 3.45, 8.14
18. Two triangles have equal bases. The altitude of VI. Spherical Trigonometry
one triangle is 3cm more than its base while the
altitude of the other is 3 cm less than its base. The study of properties of spherical triangles and their
Find the length of the longer altitude if the areas measurements is called spherical trigonometry.
of the triangle differ by 21 square centimeters.
a. 10 c. 14 Conversion Factors
b. 20 d. 15
1 minute of arc = 1 nautical mile
1 nautical mile = 6080 ft. = 1.1516 statute mile
V. Trigonometric Identities 1 statute mile = 5280 ft.
1 knot = 1 nautical mile per hour
Reciprocal:
Spherical Triangle
1 1 1
sin 𝑢 = cos 𝑢 = tan 𝑢 = -triangle enclosed by arcs of three great circles of a
csc 𝑢 sec 𝑢 cot 𝑢
sphere

Properties of Spherical Triangle

Quotient:
sin 𝑢 1. Sum of 3 vertex angles:
tan 𝑢 = A + B + C > 180°
cos 𝑢
A + B + C < 540°
Pythagorean: 2. Sum of any 2 sides
𝑠𝑖𝑛2 𝑢 + 𝑐𝑜𝑠 2 𝑢 = 1 b+c>a
a+c>b
Addition & Subtraction: a+b>c
sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑢 ± cos 𝑢 sin 𝑢 3. Sum of 3 sides
0° < a + b+ c < 360°
cos(𝑢 ± 𝑣) = cos 𝑢 cos 𝑣 ∓ sin 𝑢 sin 𝑣
tan 𝑢 ± tan 𝑣 4. Spherical Excess
tan(𝑢 ± 𝑣) = E = (A + B + C) – 180°
1 ∓ tan 𝑢 tan 𝑣 5. Spherical Defect
Double Angle: D = 360° - ( a + b + c)
sin 2𝑢 = 2 sin 𝑢 cos 𝑢
QUESTIONS:
cos 2𝑢 = 2 𝑐𝑜𝑠 2 𝑢 − 1
2 tan 𝑢 24. A spherical triangle ABC has sides a = 50°, c =
tan 2𝑢 = 80° and an angle C = 90°. Find the third side b of
1 − 𝑡𝑎𝑛2 𝑢 the triangle in degrees.
Inverse Trigonometric Functions: a. 75.33° c. 74.33°
b. 77.25° d. 73.44°
Inverse sine function 25. Given an isosceles triangle with angle A = B = 64°
𝑦 = 𝑎𝑟𝑐 sin 𝑥 iff sin y = x and side b = 81°. What is the value of angle C?
a. 144°26’ c. 120°15’
b. 135°10’ d. 150°25’
Inverse cosine function
𝑦 = 𝑎𝑟𝑐 cos 𝑥 iff cos y = x
Solution to Right Spherical Triangle b. 45.65 m2 d. 69.02 m2
27. If Greenwich Mean Time (GMT) is 10 AM, what is
Napier’s Rules the time at a place located 120° West longitude?
1. The sine of any middle part is equal to the product c. 144°26’ c. 120°15’
of the cosines of its opposite parts.’ d. 135°10’ d. 150°25’
__ __
sin a = cos A cos c
__ __
sin b = cos B cos c
__
sin c = cos a cos b
__ __
sin A = cos a cos B
__ __
sin B = cos A cos b

2. The sine of any middle part is equal to the product

of the tangents of its adjacent parts.
__
sin a = tan b tan B
__
sin b = tan a tan A
__ __ __
sin c = tan A tan B
__ __
sin A = tan b tan c
__ __
sin B = tan a tan c

Area of Spherical Triangle

𝜋 𝑟2𝐸
𝐴=
180

Where: E = spherical excess

r = radius of sphere

If given three sides:

𝐸 𝑠 𝑠−𝑎 𝑠−𝑏 𝑠−𝑐
tan = √tan tan tan tan
4 2 2 2 2
𝑎+𝑏+𝑐
𝑠=
2

If given three angles:

E = A + B + C - 180°

Laws of Sines
sin 𝑎 sin 𝑏 sin 𝑐
= =
sin 𝐴 sin 𝐵 sin 𝐶

Laws of Cosines for Sides

cos 𝑎 = cos 𝑏 cos 𝑐 + sin 𝑏 sin 𝑐 cos 𝐴
cos 𝑏 = cos 𝑎 cos 𝑐 + sin 𝑎 sin 𝑐 cos 𝐵
cos 𝑐 = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏 cos 𝐶

Laws of Cosines for Angles

cos 𝐴 = − cos 𝐵 cos 𝐶 + sin 𝐵 sin 𝐶 cos 𝑎
cos 𝐵 = − cos 𝐴 cos 𝐶 + sin 𝐴 sin 𝐶 cos 𝑏
cos 𝐶 = − cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 cos 𝑐

QUESTIONS:
26. Compute the area of bi-rectangular spherical
triangle having an angle of 60° and a radius of
8m.
a. 76.56 m2 c. 56.34 m2