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SPLBE2016

PLANE TRIGONOMETRY LECTURE NOTES

PLANE TRIGONOMETRY

COURSE OUTLINE Types of Triangles Right Triangle 1. Area of a Triangle 2. Pythagorean Theorem 3. Trigonometric Functions 4. Trigonometric Identities C. Obtuse Triangle 1. Area of a Triangle 2. Sine Law 3. Cosine Law 4. Law of Tangents D. Area of a Triangle E. Properties and Elements of a Triangle F. Angles A. B.

A. TYPES OF TRIANGLE

B. RIGHT TRIANGLE

C. OBLIQUE TRIANGLE SINE LAW π π π = = πππ π΄ πππ π΅ πππ πΆ COSINE LAW a2 = b2 + c2 β 2bc CosA b2 = a2 + c2 β 2ac CosB

Area of a Right Triangle A = Β½ bh A = Β½ ab sinC = Β½ cb sinA A = βs(s β a)(s β b)(s β c) where s = Β½ (a+b+c) A=

π2

π πππ΅ π πππΆ 2 π πππ΄

c2 = a2 + b2 β 2ab CosC

ACUTE ANGLE is an angle < 90o. RIGHT ANGLE is an angle = 90o. OBTUSE ANGLE is an angle > 90o.

Pythagorean Theorem a2 + b2 = c2

STRAIGHT ANGLE is an angle = 180o.

Trigonometric Functions

COMPLEMENTARY ANGLES are angles whose sum is 90o.

ππππππ ππ‘π

Sin ΞΈ = βπ¦πππ‘πππ’π π ππππππππ‘ βπ¦πππ‘πππ’π π ππππππ ππ‘π π πππ Tan ΞΈ = ππππππππ‘ = πππ π ππππππππ‘ πππ π Cot ΞΈ = πππππ ππ‘π = π πππ βπ¦πππ‘πππ’π π 1 Sec ΞΈ = ππππππππ‘ = πππ π βπ¦πππ‘πππ’π π 1 Csc ΞΈ = πππππ ππ‘π = π πππ

Cos ΞΈ =

REFLEX ANGLE is an angle >180o.

SUPLEMENTARY ANGLES are angles whose sum is 180o. EXPLEMENTARY ANGLES are angles whose sum is 360o. UNITS OF ANGLES. 90o = Ο/2 radians = 100grads = 1600mils

IA | Page 1 of 3

SPLBE2016

PLANE TRIGONOMETRY LECTURE NOTES

TRIGONOMETRIC IDENTITIES Sum of Two Angles Sin (A+B) = SinA CosB + CosA SinB Cos (A+B) = CosA CosB β SinA SinB Tan (A+B) =

TanA + TanB 1 β TanA TanB

Difference of Two Angles Sin (A-B) = SinA CosB β CosA SinB Cos (A-B) = CosA CosB + SinA SinB TanA β TanB 1 + TanA TanB

Tan (A-B) =

Double Angles Sin 2A = 2 SinA CosA Cos 2A = Cos2A β Sin2A 2 TanA

Tan 2A = 1β Tan2 A Half Angles 1βπΆππ π΄ 2

Sin A/2 = β

1+πΆππ π΄ 2

Cos A/2 = β

1βπΆππ π΄

Tan A/2 = β1+πΆππ π΄ QUADRILATERALS

ELEMENTS OF A TRIANGLE Median, line drawn from the vertex of a triangle to the midpoint of the opposite side. Centroid, the common point where all Medians of the triangle intersect. Altitude, a perpendicular line drawn from the vertex of a triangle to the opposite side. Orthocenter, the point where all Altitudes of the triangle intersect. Angle Bisector, line drawn from one vertex of a triangle to the opposite side by bisecting the included angle between two sides. Incenter, the point where all Angle Bisectors of a triangle intersects.

ALTITUDE OF A TRIANGLE is a perpendicular from any vertex to the opposite side. ANGLE is the opening between two straight lines drawn from a point. APOTHEM of a polygon is the radius of its inscribed circle. CENTER OF POLYGON of a regular polygon is the common center of the inscribed and circumscribed circle. CONCURRENT LINES are three or more line which have one point in common. DIAGONAL of a polygon is a line connecting 2 non-adjacent vertices of the polygon.

Any Quadrilateral A = Β½ d1 d2 sin ΞΈ A = β(s β a)(s β b)(s β c)(s β d) β abcd cos2 ΞΈ Where s = (a+b+c+d)/2 ΞΈ = (A+C)/2 or (B+D)/2 Cyclic Quadrilateral A = β(s β a)(s β b)(s β c)(s β d) A+C = B+D = 180O Ptolemyβs Theorem: ac + bd = d1 d2

IA | Page 2 of 3

SPLBE2016

PLANE TRIGONOMETRY LECTURE NOTES

SAMPLE PROBLEMS 1. 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 48o. From the same point, the angle of depression to the foot of tower B is 26 o. What is the height of tower B.

2. Two sides of a triangle are 10cm and 25cm, respectively. The perimeter maybeβ¦

3. The area of a triangle is 40 cm2 and two of its sides measure 10cm & 16cm, respectively. What is the length of the 3rd side?

13. If tan A = y/x, then cos (90oβA) is equal to A. y / (x2 + y2)

B. y / (x2 + y2)0.5

A. 25.16 cm

B. 9.02 cm

C. x / (x2 + y2)

D. x / (x2 + y2)0.5

C. 12.34 cm

D. 15.78 cm

14. If A, B and C are interior angles of a triangle and tanA + tanB + tanC = x, what is tanA tanB tanC equal to?

4. The angles of a triangle are in ratio 2:3:7 and its perimeter is 20cm. Find the longest side. 5. A triangle has an area of 9.92cm2 and its perimeter is 15cm. Find the distance from the point of intersection of its angle bisectors to one side. 6. A pole cast a shadow 15m long when the angle of elevatior of the sun is 63o. If the pole leans 15o from the vertical toward the sun, determine the length of the pole. 7. A boat left Port O and sailed S46oE at 50mph. Two hours later, another boat left the same port and sailed N28oE at 40mph. What is the bearing of the 1st boat from the 2nd boat two hours after the 2nd boat left port O? 8. Two boys A and B started from the same point on a circular trach 20m in radius and ran the same rate of 2m/s; A running toward the center of the circular track and B along the circumference. Haw far apart are they after 8 seconds? (from CE Board Exam) 9. From the top of a tower 100m high, two point A and B in the plane of its base have angles of depression of 15o and 12o, respectively. The horizontal angle subtended by points A and B at the foot of the tower is 48o. Find the distance between two points. 10. A triangle ABC has sides a=6cm, b=8cm and c=9cm. Find the length of the median drawn from vertex A to side BC. 11. If arcsin(3x-4y) = 1.5708 and arcos(x-y) = 1.0472, what is the value of x? (from CE Board Exam)

A. 72cm

B. 69cm

C. 39cm

D. 50cm

A. x2 β 1

B. x

C. 1/x

D. none of these

15. A quadrilateral ABCD is inscribed in a circle with the extensions of sides AB and CD meeting at an external point O. The angle of intersection (at O) of these two sides is 30o. If AB=5m, CD=8m and AO=11m, find the area of the quadrilateral. 16. When Mt. Pinatubo showed signs of activity, PHILVOCS set up stations to monitor the volcano. Two such station were A and B, 7kms apart and lying on the same horizontal plane as the base of the volcano with B nearer to the volcano. The angle of elevation at the top of Mt. Pinatubo as observed from A was 8o. At the height of Mt. Pinatuboβs fury, the vertical smoke from its crater subtended a vertical angle of 64o at each station. Find the height of Mt. Pinatubo and the vertical smoke. Assume that stations A and B and the vertical smoke lie to the same vertical plane. 17. 4 condominium buildings A, B, C and D in Bonifacio Global City are equidistant from a shopping mall M. Condo A and D are collinear with the shopping mall. If AB=400m, BC=600m and CD=800m, find the distance from the mall to each of the condominium. 18. From a point inside an equilateral triangle, the distances to the vertices are 4cm, 5cm and 6cm, respectively. Find the perimeter of the triangle.

12. If sin3A = cos12B, then A+4B is equal to A. 20o

B. 30o

C. 40o

D. 50o

IA | Page 3 of 3

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PLANE TRIGONOMETRY LECTURE NOTES

PLANE TRIGONOMETRY

COURSE OUTLINE Types of Triangles Right Triangle 1. Area of a Triangle 2. Pythagorean Theorem 3. Trigonometric Functions 4. Trigonometric Identities C. Obtuse Triangle 1. Area of a Triangle 2. Sine Law 3. Cosine Law 4. Law of Tangents D. Area of a Triangle E. Properties and Elements of a Triangle F. Angles A. B.

A. TYPES OF TRIANGLE

B. RIGHT TRIANGLE

C. OBLIQUE TRIANGLE SINE LAW π π π = = πππ π΄ πππ π΅ πππ πΆ COSINE LAW a2 = b2 + c2 β 2bc CosA b2 = a2 + c2 β 2ac CosB

Area of a Right Triangle A = Β½ bh A = Β½ ab sinC = Β½ cb sinA A = βs(s β a)(s β b)(s β c) where s = Β½ (a+b+c) A=

π2

π πππ΅ π πππΆ 2 π πππ΄

c2 = a2 + b2 β 2ab CosC

ACUTE ANGLE is an angle < 90o. RIGHT ANGLE is an angle = 90o. OBTUSE ANGLE is an angle > 90o.

Pythagorean Theorem a2 + b2 = c2

STRAIGHT ANGLE is an angle = 180o.

Trigonometric Functions

COMPLEMENTARY ANGLES are angles whose sum is 90o.

ππππππ ππ‘π

Sin ΞΈ = βπ¦πππ‘πππ’π π ππππππππ‘ βπ¦πππ‘πππ’π π ππππππ ππ‘π π πππ Tan ΞΈ = ππππππππ‘ = πππ π ππππππππ‘ πππ π Cot ΞΈ = πππππ ππ‘π = π πππ βπ¦πππ‘πππ’π π 1 Sec ΞΈ = ππππππππ‘ = πππ π βπ¦πππ‘πππ’π π 1 Csc ΞΈ = πππππ ππ‘π = π πππ

Cos ΞΈ =

REFLEX ANGLE is an angle >180o.

SUPLEMENTARY ANGLES are angles whose sum is 180o. EXPLEMENTARY ANGLES are angles whose sum is 360o. UNITS OF ANGLES. 90o = Ο/2 radians = 100grads = 1600mils

IA | Page 1 of 3

SPLBE2016

PLANE TRIGONOMETRY LECTURE NOTES

TRIGONOMETRIC IDENTITIES Sum of Two Angles Sin (A+B) = SinA CosB + CosA SinB Cos (A+B) = CosA CosB β SinA SinB Tan (A+B) =

TanA + TanB 1 β TanA TanB

Difference of Two Angles Sin (A-B) = SinA CosB β CosA SinB Cos (A-B) = CosA CosB + SinA SinB TanA β TanB 1 + TanA TanB

Tan (A-B) =

Double Angles Sin 2A = 2 SinA CosA Cos 2A = Cos2A β Sin2A 2 TanA

Tan 2A = 1β Tan2 A Half Angles 1βπΆππ π΄ 2

Sin A/2 = β

1+πΆππ π΄ 2

Cos A/2 = β

1βπΆππ π΄

Tan A/2 = β1+πΆππ π΄ QUADRILATERALS

ELEMENTS OF A TRIANGLE Median, line drawn from the vertex of a triangle to the midpoint of the opposite side. Centroid, the common point where all Medians of the triangle intersect. Altitude, a perpendicular line drawn from the vertex of a triangle to the opposite side. Orthocenter, the point where all Altitudes of the triangle intersect. Angle Bisector, line drawn from one vertex of a triangle to the opposite side by bisecting the included angle between two sides. Incenter, the point where all Angle Bisectors of a triangle intersects.

ALTITUDE OF A TRIANGLE is a perpendicular from any vertex to the opposite side. ANGLE is the opening between two straight lines drawn from a point. APOTHEM of a polygon is the radius of its inscribed circle. CENTER OF POLYGON of a regular polygon is the common center of the inscribed and circumscribed circle. CONCURRENT LINES are three or more line which have one point in common. DIAGONAL of a polygon is a line connecting 2 non-adjacent vertices of the polygon.

Any Quadrilateral A = Β½ d1 d2 sin ΞΈ A = β(s β a)(s β b)(s β c)(s β d) β abcd cos2 ΞΈ Where s = (a+b+c+d)/2 ΞΈ = (A+C)/2 or (B+D)/2 Cyclic Quadrilateral A = β(s β a)(s β b)(s β c)(s β d) A+C = B+D = 180O Ptolemyβs Theorem: ac + bd = d1 d2

IA | Page 2 of 3

SPLBE2016

PLANE TRIGONOMETRY LECTURE NOTES

SAMPLE PROBLEMS 1. 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 48o. From the same point, the angle of depression to the foot of tower B is 26 o. What is the height of tower B.

2. Two sides of a triangle are 10cm and 25cm, respectively. The perimeter maybeβ¦

3. The area of a triangle is 40 cm2 and two of its sides measure 10cm & 16cm, respectively. What is the length of the 3rd side?

13. If tan A = y/x, then cos (90oβA) is equal to A. y / (x2 + y2)

B. y / (x2 + y2)0.5

A. 25.16 cm

B. 9.02 cm

C. x / (x2 + y2)

D. x / (x2 + y2)0.5

C. 12.34 cm

D. 15.78 cm

14. If A, B and C are interior angles of a triangle and tanA + tanB + tanC = x, what is tanA tanB tanC equal to?

4. The angles of a triangle are in ratio 2:3:7 and its perimeter is 20cm. Find the longest side. 5. A triangle has an area of 9.92cm2 and its perimeter is 15cm. Find the distance from the point of intersection of its angle bisectors to one side. 6. A pole cast a shadow 15m long when the angle of elevatior of the sun is 63o. If the pole leans 15o from the vertical toward the sun, determine the length of the pole. 7. A boat left Port O and sailed S46oE at 50mph. Two hours later, another boat left the same port and sailed N28oE at 40mph. What is the bearing of the 1st boat from the 2nd boat two hours after the 2nd boat left port O? 8. Two boys A and B started from the same point on a circular trach 20m in radius and ran the same rate of 2m/s; A running toward the center of the circular track and B along the circumference. Haw far apart are they after 8 seconds? (from CE Board Exam) 9. From the top of a tower 100m high, two point A and B in the plane of its base have angles of depression of 15o and 12o, respectively. The horizontal angle subtended by points A and B at the foot of the tower is 48o. Find the distance between two points. 10. A triangle ABC has sides a=6cm, b=8cm and c=9cm. Find the length of the median drawn from vertex A to side BC. 11. If arcsin(3x-4y) = 1.5708 and arcos(x-y) = 1.0472, what is the value of x? (from CE Board Exam)

A. 72cm

B. 69cm

C. 39cm

D. 50cm

A. x2 β 1

B. x

C. 1/x

D. none of these

15. A quadrilateral ABCD is inscribed in a circle with the extensions of sides AB and CD meeting at an external point O. The angle of intersection (at O) of these two sides is 30o. If AB=5m, CD=8m and AO=11m, find the area of the quadrilateral. 16. When Mt. Pinatubo showed signs of activity, PHILVOCS set up stations to monitor the volcano. Two such station were A and B, 7kms apart and lying on the same horizontal plane as the base of the volcano with B nearer to the volcano. The angle of elevation at the top of Mt. Pinatubo as observed from A was 8o. At the height of Mt. Pinatuboβs fury, the vertical smoke from its crater subtended a vertical angle of 64o at each station. Find the height of Mt. Pinatubo and the vertical smoke. Assume that stations A and B and the vertical smoke lie to the same vertical plane. 17. 4 condominium buildings A, B, C and D in Bonifacio Global City are equidistant from a shopping mall M. Condo A and D are collinear with the shopping mall. If AB=400m, BC=600m and CD=800m, find the distance from the mall to each of the condominium. 18. From a point inside an equilateral triangle, the distances to the vertices are 4cm, 5cm and 6cm, respectively. Find the perimeter of the triangle.

12. If sin3A = cos12B, then A+4B is equal to A. 20o

B. 30o

C. 40o

D. 50o

IA | Page 3 of 3

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