Download Sin, Cos and Tan Functions Opposite Side

Physics/Conceptual Physics
Overview:
Geometry
Graphing
Trigonometry
Triangles
• Triangles are three sided shapes.
• Sum of the interior angles: The three interior angles
of any triangle add up to 180°.
• The sum of the lengths of any two sides of a
triangle always exceeds the length of the third side,
a principle known as the triangle inequality.
Triangles
In the figure below
x = ?.
Triangles
In the figure below
x + 50 + 100 = 180
so x = 30.
Triangles
Area formula: The general formula for the area of
a triangle is always the same.
• The formula is:
• Area of Triangle= (1/2)(base)(height)
Triangles
What is the area of the following triangle:
Triangles
What is the area of the following triangle:
Area = b x h = 6 cm x 9 cm = 27 cm2
2
2
Assessment Question 1
All of the following are
true about the following
triangle EXCEPT:
A. A Sum of the interior angles: The three interior angles
of any triangle add up to 180°.
B. Area of Triangle= (1/2)(base)(height)
C. The sum of the lengths of any two sides of a triangle
always exceeds the length of the third side.
D. x = 90o, y = 2 m and the area is 15.2 m2
E. x = 60o, y = 6 m and the area is 32.5 m2
Triangles
• Triangles can be classified by:
• Sides
• Angles
Triangles
• Triangles can be classified by their sides:
• Scalene - no congruent sides.
• Isosceles - two congruent sides.
(DF = FE)
• Equilateral - three congruent
sides.
Triangles
• Triangles can be classified by their angles:
• Acute - all angles measure less than 90º.
• Right - one angle measures exactly 90º.
• Obtuse - one angle measures more than 90º.
Triangles
• Equiangular - all angles measure the same. (60º)
This is the same as the equilateral triangle.
• Similar triangles: triangles that have the same
shape
Assessment Question 2
All of the following are
true about the following
triangle EXCEPT:
A.
B.
C.
D.
E.
Triangles that can be classified as Scalene - no congruent sides;
Isosceles - two congruent sides; Equilateral - three congruent sides.
Triangles can be classified as Acute - all angles measure less than
90º; Right - one angle measures exactly 90º; Obtuse - one angle
measures more than 90º;
This is an equilateral acute Isosceles triangle
This is an obtuse triangle
This is a scalene triangle
Quadrilaterals
• Quadrilaterals are four sided shapes like:
•
•
•
•
•
Square
Rectangle
Rhombus
Trapezoid
Parallelogram
Quadrilaterals
• Trapezoids:
• A trapezoid is a four-sided figure with one pair
of parallel sides and one pair of nonparallel
sides.
Quadrilaterals
• Parallelograms:
• A parallelogram is a four-sided figure with two
pairs of parallel sides.
• Opposite sides and angles are equal.
• Consecutive angles add up to 180°.
• Area of Parallelogram = base x height
Quadrilaterals
• What is the area of the parallogram?
Quadrilaterals
• What is the area of the parallogram in ft2?
• Area= base x height = 6 ft x 4 ft = 24 ft2
Assessment Question 3
All of the following are
true about the following
quadrilateral EXCEPT:
A. A trapezoid is a four-sided figure with one pair of
parallel sides and one pair of nonparallel sides.
B. A parallelogram is a four-sided figure with two pairs
of parallel sides.
C. The Area of Parallelogram = base x height
D. This is a parallelogram with an area of 35 m2.
E. This is a trapezoid with an area of 18 m2.
Quadrilaterals
• Rectangles:
• A rectangle is a four-sided figure with four right
angles.
• Opposite sides are equal.
• Diagonals are equal.
Quadrilaterals
• The perimeter of a rectangle is equal to the sum
of the lengths of the four sides, which is equal
to 2(length + width).
• Area of Rectangle = length x width
Quadrilaterals
• What is the area of the following rectangle in
square miles?
Quadrilaterals
• What is the area of the following rectangle in
square miles?
• Area = length x width = 3 mi x 7 mi = 21 mi2
Quadrilaterals
• Rhombus:
• A four-sided figure with four equal sides.
• Area of a rhombus = base x height
Quadrilaterals
• Square:
• A rhombus that has each pair of adjacent sides
perpendicular.
• Area of Square = (side)2
Assessment Question 4
All of the following are
true about the following
quadrilateral EXCEPT:
A.
B.
C.
D.
E.
A rectangle is a four-sided figure with four right angles.
Area of Rectangle = length x width
A rhombus is a 4-sided figure with equal sides.
Area of a rhombus = base x height
A square is a rhombus with all sides perpendicular.
Area of
Square = (side)2
This is a rectangle with an area of 25 m2.
This is a rhombus.
Circles
• All circles are similar-they're all the same shape.
• The only difference among them is size.
•
• So you don't have to learn to recognize types or
remember names.
Circles
• Pi : π
• is a mathematical constant which represents the
ratio of any circle's circumference to its
diameter
• π = 3.14159265358979323846
• pi is a continuous decimal, it will never stop
repeating so we shorten it to
• π = 3.14
Circles
• π=
3.141592653589793238462643383279502884197
16939937510582097494459230781640628620899
86280348253421170679821480865132823066470
93844609550582231725359408128481117450284
10270193852110555964462294895493038196442
88109756659334461284756482337867831652712
01909145648566923460348610454326648213393
60726024914127372458700660631558817488152
09209628292540917153643678925903600113305
30548820466521384146951941511609...
Circles
•
•
•
•
Circumference:
Circumference is a measurement of length.
You could think of it as the perimeter:
It's the total distance around the circle.
Circles: Circumference
• If the radius of the circle is r:
• Circumference = 2πr
• Since the diameter is twice the radius, you can
easily express the formula in terms of the
diameter d:
• Circumference = πd
Circles: Area
• If the radius of the circle is r:
• Area = πr2
Circles
• What is the area and circumference of the
following circle with r= 4 cm?
Circles
• What is the area and circumference of the
following circle with r = 4 cm?
• Area = πr2 = 3.14 x 4 cm2
• Area = 50 cm2
• Circumference = 2πr
• 2πr= 2 x 3.14 x 4 cm
• Circumference = 25 cm
Assessment Question 5
All of the following are
true about the following circle
A.
B.
C.
D.
E.
EXCEPT:
If the radius of the circle is r:
Circumference = 2πr
Area = πr2
Pi (π) is a ratio of any circle’s circumference to its diameter.
π = 3.14
The diameter is always greater than the circumference of a
circle.
The area of the circle is 78.5 cm2.
The circumference of the circle is 31.4 cm.
Rectangular Solids
• The rectangular solid is the official geometric
term for a box, which has six rectangular faces:
• The volume of rectangular solids is equal to the
product of the length, height and width.
• V= l x h x w
Rectangular Solids
• What is the volume of rectangular solid below in
ft3:
• V= l x h x w
Rectangular Solids
• What is the volume of rectangular solid below in
ft3:
• V= l x h x w
• V = 7’ x 4’ x 4’
• V = 112 ft3
Assessment Question 6
Calculate the volume
of the rectangular
solid:
V= l x h x w
A.
B.
C.
D.
E.
19 m3
240 m3
830 m3
70 m3
88 m3
Graphing functions
• In interpreting the results of a scientific
experiment, it is often useful to make a graph.
• If possible the function should be graphed in a
straight line.
Linear Equation
• The equation for a straight line :
y = mx + b
y is the dependent variable
x is the independent variable
m is the slope
b is the intercept with the y axis
Linear Equation
• The slope of a line is defined as the ratio of the
rate of change in y to x.
• The slope is the change in y divided by the
change in x.
Linear Equation
• The equation for the slope of a straight line :
m = slope = ∆y = y2- y1
∆x x2- x1
∆ = change in each variable
Graphing Linear Coordinates
• When graphing linear coordinates the x value is
the number indicated first and the y value is
second: (x, y)
Assessment Question 7
All of the following are true EXCEPT:
A.
B.
C.
D.
The equation for a straight line : y = mx + b
slope = m = (y2 - y1) / (x2 - x1)
The line y = 3x + 2 has a slope of 2
The slope between the points (1,2) and (2,4) is
½.
E. The line y = 6x + 7 has a y-intercept of 7
Coordinate Grids
• When setting up a coordinate grid:
• The horizontal axis is the x-axis
• The vertical axis is the y –axis
•
Coordinate Grids
• When setting up a coordinate grid:
• The x and y axis meet at (0,0)
• x values to the left of the y axis are negative
• y values below the x axis are negative
Graphing Linear Coordinates
• When graphing linear coordinates the x value is
the number indicated first and the y value is
second: (x, y)
• Graph the following coordinates:
• (2,3) and (3,2)
Graphing Coordinates
Assessment Question 8
•
All of the following
coordinates are correct
EXCEPT
A.
B.
C.
D.
E.
(1,3)
(-3,2)
(1,2)
(2,1)
(3,-3)
Sin, Cos and Tan Functions
• The trigonometric functions of angles are the
ratios of the various sides of a triangle.
Sin, Cos and Tan Functions
• Consider a right-angled triangle ABC as shown in
the figure below.
Sin, Cos and Tan Functions
• Hypotenuse: The side opposite to the right
angle in a triangle is called the hypotenuse.
Here the side AC is the hypotenuse.
Sin, Cos and Tan Functions
• Opposite Side: The side opposite to the angle in
consideration is called the opposite side. So, if
we are considering angle A, then the opposite
side is CB.
Sin, Cos and Tan Functions
• Adjacent Side (Base): The third side of the
triangle, which is one of the arms of the angle
under consideration, is called the base. If A is
the angle under consideration, then the side
• AB is the base.
Sin, Cos and Tan Functions
• For angle A (sometimes referred to as angle
CAB), the following fundamental trigonometric
functions can be defined.
•
Assessment Question 9
All of the following are correct
EXCEPT
A. The adjacent side of a triangle is adjacent to the angle
in consideration.
B. The side opposite to the right angle in a triangle is called
the hypotenuse.
C. The side opposite to the angle in consideration is called
the opposite side.
D. If angle B is a right angle than side b is the hypotenuse
E. If angle A is the angle in consideration than side a is
adjacent.
Sin, Cos and Tan Functions
• Sine of A = sin A = Opposite Side / Hypotenuse
• sin A= CB/CA = a/b
Sin, Cos and Tan Functions
• Cosine of A = cos A = Adjacent/ Hypotenuse
• cos A = AB/CA = c/b
•
Sin, Cos and Tan Functions
• Tangent of A = tan A = Opposite Side / Base
• tan A = CB/AB = a/c
•
Sin, Cos and Tan Functions
• If we know the value of an angle in a right
triangle,
• Sin, cos and tan functions will tell us the ratio of
the sides of the triangle.
Sin, Cos and Tan Functions
• If we know the length of any one side, we can
solve for the length of the other sides.
• Or if we know the ratio of any two sides of a
right triangle, we can find the value of the angle
between the sides using sin, cos and tan
functions will .
Sin, Cos and Tan Functions
• Remember SOH CAH TOA
and TSC
• SOH:
• CAH:
• TOA:
Sin = Opposite/ Hypotenuse
Cos = Adjacent/ Hypotenuse
Tan = Opposite/ Adjacent
• TSC:
Tan θ = Sin θ / Cos θ
Assessment Question 10
•
All of the following are true for functions for
angle A EXCEPT
a = 3, b = 5, c = 4
Remember SOHcahTOA
A. sin A= a/b = 3/5
B. cos A = c/b = 5/4
C. tan A = a/c = 3/4
D. cos A = c/b = 4/5
E. tan A = sin A / cos A