11.1 - Apply Trigonometric Functions to solve problems involving distance and angles

11.1- A block bordering Gillette street is a right triangle as shown in the attachment. The person begins at the intersection of Chase and Gillette streets and walks 600' along Gillette street to the intersection with white street. Then the person turns and walks forging for 455' along white street to its intersection with Chase street to the nearest degree at what angle do white and Gillette streets intersect?

This question requires the examinee to apply trigonometric functions to solve problems involving distance and angles. let ϴ represent the data between Gillette and white streets arc cosϴ=455/600 = 41°. Competency 11.1- Apply trignometric functions to solve problems involving distance and angles.

a circular prism has a d=22ft and h=10ft, what is the volume?

V= πr²h π×(11)²×10 = 3801.32³

quadrilateral

a four-sided polygon that equal 360°

pentagonal prism

a prism where the base and the top are congruent, parallel pentagons and all the other sides are rectangles

octagonal pyramid

a solid shape with an octagon as the base, 8 triangular faces that come to a point (vertex)

180 degrees

π

8/16 unit circle radians

π

4/16 of unit circle

π/2

90 degrees

π/2

3/16 of unit circle

π/3

60 degrees

π/3

2/16 of unit circle

π/4

45 degrees

π/4

1/16th of unit circle

π/6

30 degrees

π/6

csc 45°

√2

sec 45°

√2

cos 45°

√2/2

sin 45°

√2/2

cot 30°

√3

tan 60°

√3

cos 30

√3/2

sin 60°

√3/2

[cos(π+ϴ), sin(π+ϴ)]

(-a,-b)

[cos(π-ϴ), sin(π-ϴ)]

(-a,b)

11/16 unit circle coordinates

(-½, -√3/2)

5/16 unit circle coordinates

(-½, √3/2)

10/16 unit circle coordinates

(-√2/2, -√2/2)

6/16 unit circle coordinates

(-√2/2, √2/2)

9/16 unit circle coordinates

(-√3/2, -½)

7/16 unit circle coordinates

(-√3/2, ½)

12/16 unit circle coordinates

(0, -1)

4/16 circle coordinates

(0, 1)

8/16 unit circle coordinates

(-1,0)

16/16 unit circle coordinates

(1,0)

[cos(2π-ϴ), sin(2π-ϴ)]

(a, -b)

13/16 unit circle coordinates

(½, -√3/2)

3/16 circle coordinates

(½, √3/2)

14/16 unit circle coordinates

(√2/2, -√2/2)

2/16 circle coordinates

(√2/2, √2/2)

15/16 unit circle coordinates

(√3/2, -1/2)

1/16 circle coordinates

(√3/2, ½)

csc 270

-1

sec 180

-1

sin(-ϴ)

-sinϴ

tan(-ϴ)

-tanϴ

tan 5pi/36

.4663

Using your calculator, find the sine of 1 radian? Sine of 1 degree?

.841, .017

cos 270

0

cos 90°

0

cot 270

0

cot 90°

0

sin 0

0

sin 180

0

sin 360 degrees

0

tan 0°

0

tan 180

0

tan 360

0

cos 0°

1

cos 360

1

cot 45°

1

csc 90°

1

sec 0

1

sec 360

1

sin 90°

1

tan 45°

1

arc and angle

1 radian

How do find the length of an edge?

1. Determine the side length when you calculate surface area. 2. solve for e, v=e³

cos 3720 degrees

1. Remove full rotations of 360° until angle is between 0° and 360°, cos(120) 2. Apply the reference <, make the expression negative because cosine is negative in the 2nd quadrant, - cos(60) = -1/2

09Find the measure of one of the acute angles in a right triangle given the length of one of its sides

1. Sin of < is opposite/hypotenuse 2. Set up the equation to solve for the hypotenuse: c=a/sinA 3. c=120cm/sin50 4. c=120/.7660= 156.65cm Competency 11.1- Find the measure of one of the acute angles in a right triangle given the length of one of its sides.

Diagnostic #2- find the distance across a large pond a surveyor took measurements tip measurements in the figure show use these measurements to determine how far it is across the lake...

1. Tan = opposite side/ adjacent side 2. Tan 50= AB/BC, AB/80 3. Tan50(80)=x= 95.2ft Competency 11.1 - Apply triangle trigonometry to solve such problems as finding the measure of one of the acute angles in a right triangle given the length of 1 of its sides.

Diagnostic 3&4 -recreation room is in the shape of an isosceles triangle with the vertex angle measuring 90ﾟ in a base 4√2 m. Please view attached.. Find the perimeter and area of the room..

1. Triangle is isosceles, AB= BC= 4√2 2. AC²= AB²+BC²=> AC²= (4√2)²+(4√2)²= 32+32 =64 3. Perimeter = 4√2+4√2+8=(16√2+8)m= 30.627m 4. Area=½×base×height =½(4√2)(4√2)= 16m² Competency 11.1- Apply trig functions to solve problems involving distance and angles.

cos 60°

1/2

sin 30°

1/2

surface area of a pyramid

1/2Pl+B

Lateral Area of a pyramid

1/2pl

cot 60°

1/√3

tan 30°

1/√3

How much does a wheel with radius 1 foot rotate if it travels 1000 feet along a road? Give the answer in radians and also in degrees.

1000 radians Degrees - 1000/2∏ =159.55 rotations, 159.55x360 = 57438

15/16 unit circle radians

11π/6

5/16 unit circle degrees

120°

convert 120 degrees to radians

120×pi/180

6/16 unit circle degrees

135°

a circular prism has a r=8km and h=7km, what is the volume?

1407.4 km³

7/16 unit circle degrees

150°

a rectangular pyramid measuring 4 in in 9 in along the base with slant heights of 10.1 in and 9.2 in respectively.

159.2 in²

8/16 unit circle degrees

180°

csc 30°

2

sec 60°

2

csc 23 degrees

2.55

csc 60°

2/√3

sec 30°

2/√3

9/16 unit circle degrees

210°

10/16 unit circle degrees

225°

11/16 unit circle degrees

240°

12/16 unit circle degrees

270°

sin2ϴ

2sinϴcosϴ

tan2ϴ

2tanϴ/1-tan²ϴ

16/16 unit circle radians

2π

360 degrees

2π

5/16 unit circle radians

2π/3

Surface area of a cylinder

2πrh+2πr²

convert 200°

3.49 radians

13/16 unit circle degrees

300°

a circular prism has a r=5in and h=4in, what is the volume?

314.2in³

14/16 unit circle degrees

315°

15/16 unit circle degrees

330°

6 sides and side length= 6 = perimeter?

36

sinϴ=⅗

36.86 degrees, Step 1 = take the inverse sine of both sides of the equation. Step 2 = sine is also positive in the 2nd quadrant, 180-36.869 = 143.13° Step 3 = the period is 360° or 2pi

16/16 unit circle degrees

360°

12/16 unit circle radians

3π/2

270 degrees

3π/2

6/16 unit circle radians

3π/4

4/sin 41 = x/sin 49

4.6

11/16 unit circle radians

4π/3

13/16 unit circle radians

5π/3

10/16 unit circle radians

5π/4

7/16 unit circle radians

5π/6

cosϴ= 4.4/11

66.4 degrees

14/16 unit circle radians

7π/4

9/16 unit circle radians

7π/6

hexagonal prism

8 faces, 18 edges, 12 vertices

convert 4/9 pi radians

80 degrees

convert 1.4 radians

80.20 degrees

<C=90° in a right triangle

<A=48 and <B=42

<A= 51° in a right triangle

<B=39 and <C=90

<A= 40° in a right triangle

<B=50 and <B=90

<A= 28° in a right triangle

<B=62 and <C=90

triangular prism

A prism that has bases that are triangles

triangular pyramid

A pyramid in which all 4 faces are triangles and has 1 vertex

square pyramid

A three dimensional shape with a square base, 4 triangular faces, and one vertex at the top

Competency 11.1

Apply Trigonometric Functions to solve problems involving distance and angles

A central angle in a circle of radius 2 units cuts off an arc 5 units long. What is the radian measure of this angle?

By definition, the radian measure if 5/2

A tunnel from point a to point B runs through a mountain attached. Which of the following is a length of the tunnel to the nearest meter?

Competency 11.1 this question requires the examinee to apply trignometric functions to solve problems involving distance and angles. By the law of cosines: c²=a²+b²-2abcosC c=√115²+165²-2(115)(165)cos74°=173.2≅173 meters

f(x) = x⁶-x²+7, is the function, even, odd or neither?

F(-x) = (-x)⁶-(-x)²+7 Since x⁶+x²+7 is an even function, the function is even.

If f(x) is any function, show that G(x) = ½(f(x)+f(-x)) Is an even function, and that H(x)=1/2(f(x)-f(-x)) Is an odd function. Use the results to show that every function can be written as the sum of an even and odd function.

Let's consider any function f(x). We can write f(x) as the sum of an even and an odd function as follows: F(x)= [1/2 f(x)+f(-x))]+[1/2f(x)-f(-x))] = g(x)+h(x) As explanation: As g(x) is an even function and h(x) is an odd function. Therefore, every function can be written as the sum of an even and an odd function. This is known as the even-odd decomposition function.

(cosx, siny)

Q1 (a,b)

(-cosx, siny)

Q2

(-cosx, -siny)

Q3

(cosx, -siny)

Q4

If a is an angle between 0 and ∏/2 (in radian measure), which is bigger: sin a or cos (∏ /2 - a)?

Sin angles: 30, 45, 60, 90 Cos (90-a): 60, 45, 30, 0, sin

Is sin 500 (in radian measure) a positive or a negative number?

Sin500/2∏ - is about 79 rotations + .57 more this falls within the ½ and ¾ rotation; in the third quadrant, sine is negative which is where the rotation will end.

let us take an angle whose radian measure is 1. using the picture below, prove that its degree-measure is less than 60°. (In fact, an angle of radian measure 1 is approximately 57 degrees.)

Since radian measure is defined as the ratio of the length of the arc to the radius of the circle, [since it fits into a rectangle] an angle with radian measure1 corresponds to an arc on the unit circle that is equal in length to the circumference of the circle, which is 2∏. In degrees, a full circle is 360 degrees. Therefore, an angle with radian measure 1 would correspond to a fraction of a full circle equal to 1/2∏ which is approximately .159. To convert this fraction of a circle to degrees, we can multiply it by 360 degrees. .159 x 360˚ = 57.24˚

Determine the rest of the angles and sides given: <B= 28.81, a= 14, b=7.7

Step 1: The law signs produces an ambiguous angle result. Therefore, there are 2 angles that will correctly solve the equation. Step 2: the law of sines states That for the angles of a non right triangle, each angle of the triangle has the same ratio of angle measures to the sine value. Sin(A)/a=sin(B)/b=sin(C)/c Step 3: Substitute the known values into the law of sines to find A. sin(A)/14=sin(28.81)/7.7 Step 4: solve the equation for A. A=arcsin(.87619383) A=61.18 To find the reference angle, 2nd solution (the sine function is positive in the 1st & 2nd quadrant):...180-66.18 = 118.81 Step 5: the sum of all the angles in a triangle is 180°. 61.86+c+28.81=180 c=90 Step 6: to find side c, sin(90.00341758)/c= sin(61.18658241)/14 c=15.97

Determine <A, when a= 11.9, b=10, and c= 15.54...?

Step 1: Use the law of cosine to find the unknown angle of the triangle, given the three sides. a²=b²+c²-22bccos(A) Step 2: Solve the equation A=arccos(b²+c²-a²/2bc) Step 3: Substitute the unknown values onto the equation. A=arccos((10)²+(15.54)²-(11.9)²/2(10)(15.54) Step 4: A=49.95

Determine the rest of the angles given: <A= 17.1, b=13, c=13.06...?

Step 1: the law of sines states That for the angles of a non right triangle, each angle of the triangle has the same ratio of angle measures to the sine value. Sin(A)/a=sin(B)/b=sin(C)/c Step 2: Substitute the known values into the law of sines to find B. sin(B)/13=sin(17.1)/4 Step 3: Solve the equation for B. B=72.86, 107.13 (reference <) Step4: the sum of all angles in a triangle is 180°. 17.1+C+72.8684696=180 Step 5: Solve the equation for C. C=90.03

Determine the rest of the angles given: <A= 41.81, a= 3√5, b=6, c=9...?

Step 1: the law of sines states That for the angles of a non right triangle, each angle of the triangle has the same ratio of angle measures to the sine value. Sin(A)/a=sin(B)/b=sin(C)/c Step 2: Substitute the known values into the law of sines to find B. sin(B)/6=sin(41.81)/3√5 Step 3: Solve the equation for B. B=36.604 Step4: the sum of all angles in a triangle is 180°. 41.81+C+36.60=180 Step 5: Solve the equation for C. C=101.56

Solve the equation for x where 0<x<360 degrees. sinx = 0

The exact value of arcsin(0) is 0. x=0 The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from π -0 to find the solution in the second quadrant. Subtract 0 from π. x=π sin(x)=±√1/2 Simplify ±√1/2.

In a circle of radius 1, what is the length of an arc cut off by a central angle of 2 radians?

a) 2 = ϴ/360x2∏x1 720/6.28 = ϴ = 115 degrees

In a circle of radius 3, what is the length of an arc cut off by a central angle of 2 radians?

a) 2 = ϴ/360x2∏x3 720/18.84 = ϴ = 38.21 degrees

Express the following functions as the sum of an even and odd function: f(x)=sinx + cosx

a. F(x)= sinx + cosx F(-x)= sin(-x) + cos(-x) +[f(x)= sin(x)+cos(x)]= 0 Given, f(x)=sinx+cosx f(−x)=sin(−x)+cos(−x)=−sinx+cosx ∴ f(x)=f(−x) Hence neither even nor odd.

b=29.3 in a right triangle

a=15.6 and c=33.2

c=14 mi a right triangle

a=5.2 and b=13

Properties of Quadrilaterals: square and rhombus

all sides are equal

convert 120° to radians

approx 2.09 radians

law of cosines formula

a²=b²+c²-2bcCosA

sin(37)/b=sin(53)/11

b= 8.28090455

a=3 in a right triangle

b=4 and c=5

Solve the equation for x where 0<x<360 degrees. cos^2x= ¾

cos(x)=±i√3/2 is undefined.

cos2ϴ

cos²ϴ-sin²ϴ

cos(-ϴ)

cosϴ

f(x)= sin²x+cos²x, is the function, even, odd or neither...?

f(-x)= 1 even

131°, positive or negative...

negative

cos 180

negative 1

sin 270

negative 1

Express the following functions as the sum of an even and odd function: f(x) = 2^x

neither, A function is even if f(-x) = f(x) f(-x) = 2^-x ≠2^x, this function is not even A function is odd if f(-x) = -f(x) Multiply -1 by 2^-x -f(x)= -2^-x the function is not odd If you add them together 2^-x+-2^-x we get zero again.

tan^-1(x)=1

no solutions that make the equation true

f(x) = x^3 - sinx, is the function, even, odd or neither...?

odd

To convert degrees to radians multiply by

pi/180

pentagonal pyramid

pyramid with a Pentagonal (five sided) base solid shape whose base is a pentagon with 5 triangular faces 6 faces, 10 edges, 6 vertices

a kite is a

quadrilateral - rhombus

Quadrilaterals in which all angles are equal:

rectangle, square

Quadrilaterals bisect each other

rectangle, square, parallelogram, and rhombus

Quadrilaterals in which opposite sides are equal:

rectangle, square, parallelogram, and rhombus

Quadrilaterals in which the sum of 2 adjacent angles equal: 180°

rectangle, square, parallelogram, and rhombus

Quadrilaterals in which opposite angles are equal:

rectangle, square, parallelogram, rhombus

Quadrilaterals are:

rectangle, square, parallelogram, rhombus, and trapezium

Quadrilaterals in which opposite sides are parallel:

rectangle, square, parallelogram, rhombus, and trapezium

If sin ∏/9 = cos a and a is acute, what is the radian measure of a?

sin ∏/9 =.34202014 34202014 x ∏/180 a=1.221

right triangle trigonometry

sine, cosine, tangent (SOH-CAH-TOA)

hexagonal pyramid

solid shape whose base is a hexagon and 6 triangular faces

Quadrilaterals bisect each other perpendicularly...

square, rhombus

square prism

the bases of the prism are square and the lateral faces are rectangles

use right triangle

trigonometry

cot 0

undefined

cot 180

undefined

cot 360

undefined

csc 180

undefined

csc 360

undefined

sec 270

undefined

sec 90°

undefined

tan -270 degrees

undefined

tan 270

undefined

tan 90

undefined

csc 0

undefined 1. Rewrite csc(0) in terms of sine and cosine, 1/sin(0) 2. the exact value of sin(0) is 0. stepb3 - undefined

Solve the equation for x where 0<x<360 degrees. cosx =1

x = arccos1, x= π/2 i. cos^2x= ¾ cos(x)=±i√3/2 is undefined.

10.8/sin57=x/sin90

x=12.8

3/sin90°=x/sin43°

x=2

sin90(x)=sin30×11

x=5.5

10.3/sin90=x/sin37

x=6.2

13/sin58=x/sin32

x=8.1233015739

5/sin(40.9)=x/sin(50.1)

x=approximately 6