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