Review Innovations CE Review April 2023 – Plane Trigonometry 1

ANGLES = space between two rays that extend from a common Double Angle:
point called the vertex. sin 2A = 2 sin A cos A
Acute angle – angle less than 900
cos 2A = cos2A – sin2A
Right Angle – angle equal to 900
Obtuse Angle – angle greater than 900 2tan A
tan 2A = 2A
Straight Angle – angle equal to 180 0 1−tan
Reflex Angle – angle greater than 1800 Half-Angle:
Complementary Angles = 2 angles whose sum is 90 0
Supplementary Angles = 2 angles whose sum is 180 0 A 1 − cos A A 1 + cos A
Explementary Angles = 2 angles whose sum is 360 0 sin = ±√ cos = ±√
2 2 2 2

ANGLES OF MEASUREMENTS: A 1−cos A

tan
2
= ±√1+cos A
1 revolution = 2π rad
= 3600 Squares:
1−cos 2A 1 + cos 2A
= 400 grads sin2A = cos2 A =
2 2
= 6400 mils 1−cos 2A
tan2 A =
1 + cos 2A
RIGHT TRIANGLE FORMULAS:
Versine (versed sine): vers A = 1 – cos A
a c
sin A = csc A = Coversed sine: covers A = 1 – sin A
c a
Haversine: hav A = ½ (1 – cos A)
b c Exsecant: exsec A = sec A – 1
cos A = sec A =
c b

OBLIQUE TRIANGLE FORMULAS:

a b
tan A = cot A =
b a

Pythagorean Theorem: a2 + b2 = c2

OBLIQUE TRIANGLES FORMULAS:

Sine Law:
a b c
= =
sin A sin B sin C

Cosine Law:
Sine Law
a2 = b2 + c2 – 2bc cos A
a b c
b2 = a2 + c2 – 2ac cos B = =
c2 = a2 + b2 – 2ab cos C sin A sin B sin C
Cosine Law
FUNDAMENTAL RELATIONS: a2 = b2 + c 2 − 2bc cos A
b2 = a2 + c 2 − 2ac cos B
Reciprocal Relations: c 2 = a2 + b2 − 2ab cos C
cot A = 1 / tan A
sec A = 1 / cos A PERIOD, AMPLITUDE & FREQUENCY
csc A = 1 / sin A Period (T) = interval over which the graph of a function repeats.
Quotient Relations: Amplitude (A) = greatest distance of any point on the graph from a
sin A cos A
tan A = cot A = horizontal line which passes halfway between the maximum and
cos A sin A minimum values of the function.
Frequency (ω) = number of repetitions/cycles per unit of time or
Pythagorean Relations:
1/T.
sin2A + cos2A = 1
FUNCTION PERIOD AMPLITUDE
1 + tan2A = sec2A
1 + cot2A = csc2A y = A sin (Bx + C) + D 2π/B A
y = A cos (Bx + C) + D 2π/B A
IDENTITIES: y = A tan (Bx + C) + D π/B -
Co-function Identities:
sin (900 – θ) = cos θ
cos (900 – θ) = sin θ Sample Problems:
tan (900 – θ) = cot θ
1. If tan θ = 𝑥/3, then cos2 θ is _______.
Sum of Two Angles: a.
9
c.
3
sin (A + B) = sin A cos B + cos A sin B 𝑥 2 +9 √𝑥 2 +9
𝑥 𝑥2
b. d.
cos (A + B) = cos A cos B – sin A sin B √𝑥 2 +9 𝑥 2 +9

tan A + tan B
tan (A + B) = 2. A, B and C are the interior angles of a triangle. If
1−tan A tan B
(tanA)(tanB)(tanC) = 8.2424 and tanA + tanB = 2.5712, what is tanC?
Difference of Two Angles:
sin (A - B) = sin A cos B - cos A sin B
cos (A - B) = cos A cos B + sin A sin B 3. What is 𝑥 + 2𝑦 equal to if sin 3𝑥 = cos 6𝑦?
tan A − tan B a. 90° c. 45°
tan (A - B) =
1 + tan A tan B b. 60° d. 30°
 Review Innovations CE Review April 2023 – Plane Trigonometry 1

4. Evaluate: 11. A vertical pole is 10 m from a building. When the angle of

2 sin  cos  − cos  elevation of the sun is 450, the pole cast a shadow on the building 1
m high. Find the height of the pole.
1 − sin  + sin 2  − cos2 
a. sin θ c. cos θ
12. Points A and B 1000 m apart are plotted on a straight highway
b. tan θ d. cot θ
running East and West. From A, tower C is 32 o N of W and from B,
tower C is 26o N of E. Compute the shortest distance of tower C to
5. Two sides of a triangle are 10 cm and 25 cm respectively.
the highway.
Compute the probable perimeter of the triangle.
A. 72 cm C. 39 cm
13. The side of a mountain slopes upward at an angle of 21 0 14’. At
B. 69 cm D. 50 cm
a point on the mountain side, a mine tunnel is constructed at an
angle of 150 27’ downward from the horizontal. Find the vertical
6. Find the amplitude & period of distance to the surface of the mountain from a point 250 meters
y = sin x cos x down the tunnel.
7. Given a triangle ABC, how many possible triangle/s can be 14. From a point A at the foot of the mountain, the angle of elevation
formed for the ff conditions: AB=18 m, AC=25 m and Angle C=42 0 of the top B is 600. After ascending the mountain one mile at an
inclination of 300 to the horizon and reaching a point C, observer
8. Given two sides and an angle of a triangle ABC: AB=40 cm, finds that the angle ACB is 1350. Compute the height of the
AC=35 cm, Angle B=650. How many distinct triangle/s can be mountain.
formed?
15. The obelisk of a certain Rizal monument rises to some height
9. An observer measured the angle of elevation of the top of the above its dais; the angles of elevation of the top and bottom of the
building and found it to be 26.40. After moving 74.5 m closer to the obelisk from two stations A and B on the same horizontal plane as
building, the angle of elevation of the top of the building was 52.6 0. the base of the dais are 450 and 300 respectively. The corresponding
How high is the building? horizontal angles to the common center of both dais and obelisk
from the ends of the base line AB 25 meters long are 750 and 600
10. A pole cast a shadow 15 m long when the angle of elevation of respectively. Find the height of the obelisk.
the sun is 630. If the pole leans 150 from the vertical towards the sun,
determine the length of the pole. 16. Towers A and B stand on a level ground. From the top of tower
A which is 30m high, the angle of elevation of the top of tower B is
11. From the top of a tower 100 m high, two points A and B in the 480. From the same point, the angle of depression to the foot of
plane of its base have angles of depression of 15 0 and 120 tower B is 260. What is the height of tower B?
respectively. The horizontal angle subtended by points A and B at
the foot of the tower is 480. Find the distance between the two
points.
Answer Key:
12. A boat left Port O and sailed S 460 E at 50 mph. Two hours later, 1. Amplitude = 4; Period = ; Phase Shift = -/4
another boat left the same port at 40 mph and sailed N 280 E. What 2. 3000
is the bearing of the 1st boat from the 2nd boat two hours after the 2nd 3. x = 1.00
boat left Port O? 4. 1 – 2x2
5. 10 cm
Problems for Practice: 6. 630
1. Determine the amplitude, period and phase shift of the function: 7. 2850
y = 4sin(2x+/2) 8. One Triangle
9. 21.73 m
2. If 2700≤A≤3600 and vers A = 0.5, find A. 10. 47.10 m
11. 11 m
3. If arcsin (3x-4y) = 1.5708 and arcos (x-y) = 1.0472, what is the value 12. 273.92 m
of x? 13. 160.23 m
14. 12,492.61 ft
4. Find the equivalent expression for cos(2arcsinx). 15. 10.90 m
16. 98.31 m
5. From 6:15 A.M. to 6:40 A.M. of the same day, what length of arc
is described by the minute hand of a clock 12 cm long?

6. The sum of two angles is 1600 mils and their difference is 40

grads. Find the value of the bigger angle in degrees.

7. The supplement of a certain angle is seven times its complement.

The explement of the angle is ______.
a. 285° c. 105°
b. 75° d. 15°

8. In triangle ABC, AB=11 cm, BC=5.4 cm and Angle C=154 0. How

many triangle/s can be made?

9. An airplane starts to take off when it is 240 m away from a wall

at the end of the runway which is 12 m high. If the plane takes off
at an angle of 8o, by what vertical distance will it clear the wall?

10. The angle of elevation of a point C from a point B is 290 42’. The
angle of elevation of C from another point A 31.2 m below B is 59 0
23’. How high is C from the horizontal line through A?