Similar Right Triangles Sine, Cosine and Tangent (SOH CAH TOA)

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Similar Right Triangles

Sine, Cosine and Tangent (SOH CAH TOA)

Studying properties of triangles in mathematics is called “trigonometry” which means in greek,

“triangle + knowledge”. We are learning trigonometry when we find the perimeter and area of triangles. We are learning trigonometry when we see triangle shapes represented in the real world.

In the last lesson on geometry we learned trigonometry with the Pythagorean Theorem for the sides of Right Triangles. This formula was that “the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse, written as A²+ B²= C².Right Triangles also introduce us to the Trigonometric ratios, which relate the angle measure of a right triangle to the are the ratios of the side lengths.

ine ∡A

S

=

length of side OP P OSIT E ∡A length of HY P OT ENUSE

=

^CB_AB

osine ∡A

C

=

length of side ADJACENT ∡A length of HY P OT ENUSE

=

_AB^AC

an ∡A

T

=

length of side ADJACENT ∡A length of side OP P OSIT E ∡A

=

^CB_AC

Below are three pictures of a 3-4-5 right triangle. Remember that for these special triangles the squares of the legs equal the square of the hypotenuse. Here we have: 3²+ 4²= 5²

Sine at Angle A Cosine at Angle A Tangent at Angle A

ine ∡A

S =

^CB_AB

=

₅⁴

C osine ∡A =

^AC_AB

=

³₅

T an ∡A =

_AC^CB

=

⁴₃

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In the design above, even though each triangle is larger from left to right, the proportions of the sides are the same.

an ∡A

T

=

length of side ADJACENT ∡A length of side OP P OSIT E ∡A

=

_AC^CB

=

₁₂⁵

=

₂₄¹⁰

=

₃₆¹⁵

=

₄₈²⁰

=

₆₀²⁵

As an introduction to this idea, here are four commonly used Pythagorean Triangles. The problems that follow have right triangles that differ by a scale factor but the ratio of sides, and angles are preserved with this dilation.

{3,4,5} {5,12,13} {7,24,25} {8,15,17}

in ∡B

S = ^AC_AB = ₅³ os ∡B

C = ^BC_AB = ₅⁴ an ∡B

T = _BC^AC = ³₄

in ∡B

S = ^AC_AB = ₁₃⁵ os ∡B

C = ^BC_AB = ₁₃¹² an ∡B

T = _BC^AC = ₁₂⁵

in ∡B

S = _AB^AC = ₂₄⁷ os ∡B

C = _AB^BC = ₂₅²⁴ an ∡B

T = _BC^AC = ₂₄⁷

in ∡B

S = ^AC_AB = ₁₇⁸ os ∡B

C = ^BC_AB = ₁₇¹⁵ an ∡B

T = _BC^AC = ₁₅⁸

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Example #1:

Ann and Betty are friends flying a kite together, and Betty stands in a field with the kite spool while Ann runs along a straight path of the wind direction with the kite. After 100 feet, she releases it and floats up directly overhead until the string is taut. Betty ties the string to a stake, and then says that the shape of the triangle formed {height, ground and string} is SIMILAR to the {3, 4 and 5} right triangle.

Q1: What is the length of the kite string from Betty’s stake up to the kite flying above ?

Q2: How high is the flying kite above where Ann is standing ?

Q1: Solution: Use the COSINE ratio to find the length of the HYPOTENUSE.

osine (∡B)

C

=

length of side ADJACENT ∡B length of HY P OT ENUSE

=

^BA_BC

osine (B)

C

=

¹⁰⁰_BC

45

=

¹⁰⁰_BC

BC

00 5 = 5

C

25 feet.

B = 1

Q2: Solution: Use the TANGENT ratio to find the length of the OPPOSITE side.

an (∡B)

T

=

length of side OP P OSIT E ∡B

length of side ADJACENT ∡B

=

^AC_BA

an (B)

T

=

₁₀₀^AC

34

=

₁₀₀^AC

5 feet

C

7 = A

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Example #2: A lighthouse problem.

A ship sends out a distress SOS signal (“Save Our Ship”) and the Montauk Point Light is a lighthouse that hears the call. A worker at the lighthouse sees the ship from the top light similar to using a 7-24-25 right triangle measure. The lighthouse is 51 meters above sea level.

How far away is the distress ship from the land directly below the base of the lighthouse on the rock ?

The worker stands at the top of the triangle looking down {base, height, viewing distance is in the RATIO of {7,24,25}.

The sea level is the base side of the triangle, and is OPPOSITE the corner at the top of the lighthouse where the worker stands.

The height of the Montauk Point Light is the ADJACENT side to where the worker stands.

Solution: Use the TANGENT function.

an (∡B)

T

=

length of side OP P OSIT E ∡B

length of side ADJACENT ∡B

=

^AC_BA

an (B) T

=

^BC₅₁

247

=

^AC₅₁

74 meters

C

1

₇⁶

= A

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Example #3: A Fireman’s Ladder problem.

A fireman’s ladder is 65 feet long.

Q1: How far from the base of the building should the ladder be placed to be in proportion with an

triangle ? 5, 2, 3} {base, height, hypotenuse}

{ 1 1 =

The base of ladder to wall is the OPPOSITE side of the base angle. The ladder is the HYPOTENUSE.

osine ∡A

C

=

length of side ADJACENT ∡A length of HY P OT ENUSE

=

^AB_AC

osine (A)

C

=

^AB₆₅

135

=

^AB₆₅

5 feet.

2

Q2:How much higher will the ladder reach along a building wall with proportions of a {5,12,13} triangle then with an {3,4,5} triangle shape ?

The wall height is the wall OPPOSITE the base angle as shown at right. The ladder is the HYPOTENUSE.

ine (∡A)

S

=

_AC^BC

ine (A)

S

=

^BC₈₅

1312

=

^BC₆₅

0 feet

C

6 = B

ine (∡A)

S

=

_AC^BC

ine (A)

S

=

^BC₆₅

54

=

^BC₆₅

2 feet

C

5 = B

0 2 feet higher.

6 − 5 = 8

The {ground, wall height, and ladder} is similar to the {5,13,13} right triangle shape.

Then the height is compared a {3,4,5} triangle shape.

Note: On the SHSAT there are 4 Pythagorean Triples that commonly appear in the problem sets!

{3,4,5} {5,12,13} {7,24,25} {8,15,17}

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SHSAT Lesson #21 Classwork: Sine, Cosine and Tangent

1. What is Sin(A), the ratio of the side opposite corner A to the longest side, in meters ?

IN(A)

S =_hypotenuse^opposite

A. 3m/5m B. 4m/5m C. 5m/3m D. 3m/4m E. 4m/3m

2. What is Sin(B), the ratio of the side opposite vertex B to the hypotenuse, in inches ?

IN(B)

S = _hypotenuse^opposite

A. 4in/3in B. 3in/4in C. 5in/3in D. 4in/5in E. 3in/5in

3. What is Sin(B) for this {5,12,13} right triangle, measured in feet ?

IN(B)

A. 5ft/13ft B. 12ft/13ft C. 13ft/5ft D. 5ft/12ft E. 12ft/5ft

4. What is Sin(A) for this {5,12,13} right triangle, distances measured in miles ?

IN(A)

A. 12mi/5mi B. 5mi/12mi C. 13mi/5mi D. 12mi/13mi E. 5mi/13mi

5. Find Sin(A). {7cm,24cm25cm} triangle ?

IN(A)

A. 7cm/25cm B. 24cm/25cm C. 25cm/7cm D. 7cm/24cm E. 24cm/7cm

6. Find Sin(B). {7mm,24mm,25mm} triangle.

IN(B)

A. 24mm/7mm B. 7mm/24mm C. 25mm/7mm D. 24mm/25mm E. 7mm/25mm

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SHSAT Lesson #22: Classwork (continued) 7. What is Sin(B) for this {8,15,17} right triangle ?

IN(B)

A. 8/17 B. 15/17 C. 17/8 D. 8/15 E. 15/8

8. What is Sin(A) for this {8,15,17} right triangle ?

IN(A)

A. 15/8 B. 8/15 C. 17/8 D. 15/17 E. 8/17

9. What is

2 [cos(B)]

²

− 1 ?

OS(B)

C =_hypotenuse^adjacent

A. 3/5 B. 4/5 C. 7/25 D. 16/25 E. 24/25

10. What is ^1−Cos(A)_Sin(A)

?

IN(A)

S =_hypotenuse^opposite COS(A)= _hypotenuse^adjacent

A. 5/6 B. 4/3 C. 3/5 D. 1/2 E. 1/9

11. What is

1 − 2 [sin(B)]

²

?

IN(B)

A. 5/169 B. 12/169 C. 119/169 D. 120/169 E. 1

12. What is 1 −[T an(A)]^{2T an (A)}²

?

AN(A)

T = ^opposite_adjacent

A. 1/2 B. 4/3 C. 5/3 D. 16/7 E. 24/7

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SHSAT Lesson #21: Classwork (continued) 13. What is _{1 + Cos(A)}^{Sin (A)}

?

IN(A)

A. 1/2 B. 2/3 C. 3/4 D. 4/3 E. 3/5

14. What is Cos(B)Cos(B) - Sin(B)Sin(B) ?

IN(B)

S =_hypotenuse^opposite COS(B)=_hypotenuse^adjacent

A. 163/169 B. 144/169 C. 119/169 D. 25/169 E. 12/13

15. What is 2 Sin(B)Cos(B) ?

IN(B)

S =_hypotenuse^opposite COS(B)=_hypotenuse^adjacent

A. 164/169 B. 145/169 C. 120/169 D. 25/169 E. 12/13

16. What is Cos(A)+ Sin(A) ?

IN(A)

A. 7/25 B. 24/25 C. 31/25 D. 31/24 E. 31/7

17. What is

T an(B) *

T an(A) ¹

?

AN(A)

A. 576/49 B. 49/576 C. 625/49 D. 576/625 E. 49/625

18. What is Cos(B)Cos(B) + Sin(B)Sin(B) ?

IN(A)

A. 0 B. 1 C. 2 D. 3 E. 4

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SHSAT Lesson #21: Classwork (continued)

19. Calculate:

an(A)T an(A)

?

T

−

Cos(A)Cos(A) ¹

A. 0 B. 1 C. -1 D. 2 E. -2 OS(A)

C =_hypotenuse^adjacent AN(A)

20. A surveyor walks from the base of a tree with a triangle in proportion of

{base,height,hypotenuse} and a 3-4-5 triangle as his surveying tool. After 75 feet he holds the base of the triangle horizontally and views along the hypotenuse a straight line to the top of the tree. What is the height (h) of the tree ?

A. 35 feet B. 50 feet C. 80 feet D. 100 feet E. 125 feet

AN(A)

21. A 12-foot ladder leans over a 6 foot fence which makes a triangle with a base angle of 30 degrees. What is the distance from the base of the ladder to the start of the fence, along the ground ? (Use the pythagorean theorem to solve.)

A. 3√2 feet B. 2√6 feet C. 6√2 feet D. 6√3 feet E. 1 √32 feet

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22. A ladder leans over a 6 foot fence with a base angle of 30 degrees. This forms a triangle {base,height,hypotenuse} { 6√3 16, , 2} outside the fence. The distance from the base of the ladder to the fence is one-fourth the distance from the fence to the building. Which expression shows how high the ladder will reach on the wall ?

IN(A)

23. A student says “Well, my height is 5 and a half feet and my shadow is 2 and one-fifth feet. See that 50 foot flagpole? By the Tangent Ratio, the length of its shadow is…” What is the length of the flagpole’s shadow ?

AN(A) T = ^opposite_adjacent

24. You can also use the corner ANGLE, which is the same for all similar shaped triangles, to compute the SIN, COS, or TAN of any similar shaped triangle. How far, in ground distance, is a plane flying at 30,000 feet from the airport tower if it is descending to land exactly at the tower at a 4 degree angle of depression ? (instead of the ratio of sides, put the degree angle in with the SIN, COS or TAN)

A. sin(86)= 30000/x B. cos(86)= 30000/x C. sin(4)= 30000/x D. an(4)t = 30000/x E. cos(4)= 30000/x

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25. You can also use the corner ANGLE, which is the same for all similar shaped triangles, to compute the SIN, COS, or TAN of any similar shaped triangle. Which function would you use to find how far, in flight distance along the hypotenuse, is a plane flying at 30,000 feet from the airport tower if it is descending to land exactly at the tower at a 4 degree angle of depression ?

(Grid in SIN (1), COS (2) or TAN (3))

Grid In

26. A dog leash is tied to the top of a 7 foot fence, and the dog runs as far as it can away from the fence. If the angle at the base is 70 degrees, what is the length of the dog leash?

(Sin 70 = ₁₀₀⁹⁴ = 0.94)

Grid In

27. A ramp designed with ADA (Americans with Disabilities Act) specifications has a 3 degree incline. How high will a 100 foot ramp outside a building’s front steps rise ?

(Sin 3 degrees = ₁₀₀₀⁵²³ = 0.523)

Grid In

IN(A)

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28. A lake has two ports that are on opposite shores. A person walks at a right angle away from the dock for 50 yards, then looking across to the other port, measures a 65 degree angle. How far is it across the lake ? (Tan 65 degrees = 2.145)

Grid In

AN(A) T =^opposite_adjacent

29. An 85 foot lighthouse sees a ship at a 10 degree angle of depression away from the shore. The Coast Guard ship is also seen with an angle of 70 degrees. How far is the coast guard ship from the ship in distress ? (Tan 10 = 0.176) (Tan 70 = 2.75)

Grid In

30. A 75 foot lighthouse receives a distress signal and finds a ship at 30 degrees west of due north in the water at a viewing of 7 degrees angle of depression. The coast guard ship also gets the relay, and heads toward the ship. It is located at 60 degrees east of north, and is close to shore at a 45 degree angle of depression. The lighthouse waits until the coast guard ship is at 90 degrees to view the SOS ship and then tells the SOS ship that the coast guard is on it’s way. How far is the coast guard ship from the SOS ship ? (Round to nearest foot)(Tan 7 = 0.123)(Tan 45 = 1)(Use a²+ b²= c²)

Grid In

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SHSAT Lesson #21 Homework: Sine, Cosine and Tangent

1. What is Sin(B), measurements in feet ?

IN(B)

A. 3ft/5ft B. 4ft/5ft C. 5ft/3ft D. 3ft/4ft E. 4ft/3ft

2. What is Sin(A), measured in yards ?

IN(A)

A. 4yd/3yd B. 3yd/4yd C. 5yd/3yd D. 4yd/5yd E. 3yd/5yd

3. What is Cos(A) travel measured in kilometers ?

OS(A)

A. 5km/13km B. 13km/12km C. 13km/5km D. 5km/12km E. 12km/5km

4. What is Cos(A), map distances are in miles ?

OS(A)

C =_hypotenuse^opposite

A. 24mi/7mi B. 7mi/24mi C. 25mi/7mi D. 24mi/25mi E. 7mi/25mi 5. What is Sin(B) in microscopically small units of

“nanometers” ?

IN(B)

A. 8nm/17nm B. 15nm/17nm C. 17nm/8nm D. 8nm/15nm E. 15nm/8nm

6. What is Cos(B)Cos(B) + Sin(B)Sin(B) - 1 ?

IN(A)

S = _hypotenuse^opposite COS(A)=_hypotenuse^adjacent

A. 0 B. 1 C. 2 D. 3 E. 4

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SHSAT Lesson #21: Homework (continued) 7. What is 3*Sin(A)*Csc(A) ?

osecant SC(A)

C = C = _sin(A)¹ = ^hypotenuse_opposite

A. 1 B. 2 C. 3 D. 4 E. 5

8. What is 2[cos(A)]²− 1 ?

OS(A)

A. -3/5 B. -4/5 C. -7/25 D. -16/25 E. -24/25

9. What is ^1−Cos(A)_Sin(A)

?

IN(A)

A. 5/6 B. 4/3 C. 3/4 D. 1/2 E. 1/3

10. What is 1 − 2[sin(A)] ² ?

IN(A)

A. -5/169 B. -12/169 C. -119/169 D. -120/169 E. -1

11. What is Sin(A) Cos(A) ?÷

IN(A)

A. 12/5 B. 5/12 C. 13/5 D. 12/13 E. 5/13

12. What is ^Cos(A)_Sin(A)

?

IN(A)

A. 7/25 B. 24/25 C. 25/7 D. 7/24 E. 24/7

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SHSAT Lesson #21: Homework (continued) 13. What is ^Cos(A)₁

*

Sin(A)¹

*

Sin(B)1

?

IN(A)

A. 64/17 B. 255/17 C. 17/64 D. 64/255 E. 255/64

14. What is Sin(B)Cos(A)Sin(A)Cos(B) an(B)

?

* T

AN(B) T =^opposite_adjacent

A. 15/8 B. 8/15 C. 17/8 D. 15/17 E. 8/17

15. What is 1 −[T an(A)]^{2T an (A)}²

?

AN(A)

A. -120/119 B. -119/120 C. -25/119 D. -25/12 E. -5/12

16. What is _{1 + Cos(A)}^{Sin (A)}

?

IN(A)

A. 1/2 B. 2/3 C. 3/4 D. 4/3 E. 3/5

17. What is Cos(B)Cos(B)¹

− 1 ?

OS(B)

A. 49/576 B. 144/576 C. 120/169 D. 25/169 E. 7/24

18. What is 2 Sin(B)Cos(B) ?

IN(A)

A. 168/625 B. 336/625 C. 672/625 D. 168/169 E. 31/169

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SHSAT Lesson #21: Homework (continued)

19. 4 [T an(A) *

T an(A)¹

] ?

^{A. 0}

B. 1 C. 2 D. 3 E. 4 AN(A)

20. A surveyor walks 75 feet from the base of a tree and then measures an angle of 50 degrees with his surveying tool to the top of the tree. What is the height (h) of the tree ? (Tan 50 = 1.19)

A. 89.38 B. 75 C. 62.93 D. 52.51 E. 107.11

AN(A)

21. A fireman’s ladder leans over a 6 foot fence with a base angle of 30 degrees. The distance from the base of the ladder to the fence is one quarter the distance from the fence to the building. How long is the fireman’s ladder ? (Sin 30 = 0.5 )

IN(A)

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22. A student says “Well, my height is 5 and a half feet and my shadow is 2.2 feet. See that 50 foot flagpole? By the Tangent Ratio, the length of its shadow is…” What is the length of the flagpole’s shadow ?

AN(A)

23. How far in ground distance is a plane flying at 30,000 feet above ground from the airport tower if it is descending to land exactly at the tower at a 4 degree angle of depression ? (Round to nearest mile)

Tan (4 degrees) = 0.07 and 1 mile = 5,280 feet

A. 6 miles away B. 81 miles away C. 100 miles away D. 118 miles away E. 163 miles away

AN(A)

24. A dog leash is tied to the top of a 7 foot fence, and the dog runs as far as it can away from the fence. If the angle at the top of the fence is 35 degrees, what is the length of the dog leash ? (Cos 35 deg = 0.819)

A. 8.54 feet B. 9.14 feet C. 8.90 feet D. 10.89 feet E. 15 feet

OS(A)

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25. A ramp designed with ADA (Americans with Disabilities Act) specifications has a 4 degree incline. How high will a 100 foot inclined ramp outside a building’s front steps rise ? (Sin 4 deg = 0.0698)

Grid In

IN(A)

26. A lake has two ports that are on opposite shores. A person walks at a right angle away from the dock for 50 yards, then looking across to the other port, measures a 60 degree angle. How far is it across the lake ? (Tan 60 = 1.732)

Grid In

27. A 100 foot lighthouse sees a ship at a 15 degree angle of depression. How far

is the ship from the lighthouse? (Tan 15 = 0.268) Grid In AN(A) T =^opposite_adjacent

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28. A 100 foot lighthouse sees a ship at a 10 degree angle of depression away from the shore. The Coast Guard ship is seen with an angle of depression of 70 degrees. How far is the coast guard ship from the ship in distress ? (Tan 10 = 0.176) (Tan 70 = 2.75)

Grid In

29. A 100 foot lighthouse receives a distress signal and finds a ship at 30 degrees west of due north in the water at a viewing of 7 degrees angle of depression.

The coast guard ship also gets the relay, and heads toward the ship. It is located at 60 degrees east of north, and is close to shore at a 45 degree angle of depression. The lighthouse waits until the coast guard ship is at 90 degrees to view the SOS ship and then tells the SOS ship that the coast guard is on it’s way. How far is the coast guard ship from the SOS ship ? (Round to the nearest foot) (Tan 7 = 0.123) (Tan 45 = 1) (Use a²+ b²= c²)

Grid In

30. Two condominiums are across a wide street. The first condo is 400 feet tall.

An angle of elevation of 32 degrees is the line of sight to the roof of the other building. How tall is the other building ? (Tan 32 degrees = 0.625)

Grid In

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Extra Credit Challenge Questions:

1. In triangle ABC, side a= √3 , side b= √3 and side c > 3. Let be the largest number such that thex magnitude, in degrees, of the angle opposite side exceeds Then equals:c x. x

(A) 1 50 (B) 1 20 (C) 1 05 (D) 9 0 (E) 6 0

2. If x 2 + 2 3 + 3 , x , x , ... are in geometric progression, the fourth term is:

(A) − 2 7 (B) − 1 3

₂¹

(C) 1 2 (D) 1 3

¹₂

(E) 2 7

3. Consider equations of the form x²+ b + c = 0x . How many such equations have real roots and have coefficients selected from the set of integers ?

and c

b { 2 3 4 5 61, , , , , }

(A) 0 2 (B) 9 1 (C) 8 1 (D) 7 1 (E) 6 1

4. In a ten mile race First beats Second by 2 miles and First beats Third by 4 miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third ?

(A)2 (B) 2

¹₄

(C) 2

¹₂

(D) 2

₄³

(E) 3

5. A farmer bought 749 sheep. He sold 700 of them for the price paid for the 749 sheep. The remaining 49 sheep were sold at the same price per head as the other 700. Based on the cost, the percent gain on the entire transaction is: