Angles and Triangles
Describe the differences between the following angles; acute, obtuse, straight and right.
and describe the properties of equilateral, isosceles, scalene, and right triangles:
Acute angle: measure less than 90 degrees
obtuse: angle measured to more than 90 degrees
straight: angle equal to 180 degrees
right: angle equal to 90 degrees.
Equilateral: three equal sides, three equal angles
Isosceles: two equal sides, two equal angles
Scalene: no equal sides, no equal angles
Right: has one angle of 90 degrees.
SLO: Classify a triangle according to its angle measures and side lengths
Determine the coterminal angle for 732 degrees
732 - 360-360 = 12 degrees
SLO: Determine coterminal angles for a given angle.
Determine the exact value of a trigonometric function given a special angle for the problems listed below along with which quadrant it is located in:
Sin(7π/6)
Tan(3π/4)
Cos(-π/6)
Sin(7π/6) = -½ ; 3rd quadrant
Tan(3π/4)= -1 ; 2nd quadrant
Cos(-π/6)= (√3)/2 ; 4th quadrant
SLO:
Determine the exact value of a trigonometric function given a special angle.
Determine the quadrant in which an angle lies according to the information given
Express the positive or negative for sine, cosine, and tangent in each quadrant
Quadrant 1: Sine: +, Cosine: +, Tangent: +
Quadrant 2: Sine: +, Cosine: -, Tangent:-
Quadrant 3: Sine: -, Cosine: -, Tangent: +
Quadrant 4: Sine: -, Cosine: +, Tangent:-
Simplify the following trigonometric expression:
SecθCotθ
(1/cosθ)*(cosθ/sinθ) = 1/sinθ = CSCθ
SLO: Use basic trigonometric identities to simplify trigonometric expressions.
Verify trigonometric identities by factoring, combining fractions, and multiplying an expression by “1” (a fraction in which the numerator and denominator are identical).
An <abc right triangle has side length A equal to 4, and side C equal to 5, solve for side B with the Pythagorean theorem.
B=3 ; (√5²-4²) = 3
SLO: Use the Pythagorean Theorem to determine the side lengths of a right triangle.
Convert 14π/9 into degree format and 182 degrees into radian form
14π/9 *180/π =280 degrees
182*π/180 = 91π/90 radians
SLO: Convert an angle from degree measure to radian measure and vice versa.
Maria is 6 ft tall. She is looking at the top of a building that is 40 ft tall and is standing 52 ft away from the base of the building. What is her eyesight’s angle of elevation
tan-1(34/52) = 33.1785 degrees
Explanation: Since Maria is 6 ft tall we subtract her height from the building height because she is not looking at the building from ground zero. From there we take the inverse of tangent and divide the opposite side (40-6) over adjacent (52)
SLO: Solve application problems involving inverse trigonometric functions.
Solve application problems involving right triangles.
What is the radian measurement and coordinate points for 30o ? What radian measurements and degrees it coordinates with in other quadrants of the unit circle ?
π/6
((√3)/2,1/2)
Quadrant 2: 5π/6 at 150 degrees
Quadrant 3: 7π/6 at 210 degrees
Quadrant 4: 11π/6 at 330 degree
SLO: Determine the exact value of trig functions for all ‘special angles’ (multiples of 30 or 45 degrees) on the Unit Circle.
Verify the expression in terms of sine and cosine:
(Cotθ+Tanθ)/Cscθ=1/cosθ
1/cosθ
Cotθ+Tanθ: cosθ/sinθ + sinθ/cosθ = (cos^2θ+sin^2θ)/(sinθcosθ)= 1/(sinθcosθ)
Cscθ: 1/sinθ
( 1/(sinθcosθ))/(1/sinθ) = (1/(sinθcosθ))*(sinθ/1)
1/cosθ= 1/cosθ
SLO: Verify trigonometric identities by rewriting the identity in terms of sine and cosine.
Determine the area of a right triangle with a height of 14 inches and a base of 20.
140 inches2
½(14)(20) = 140 square inches
SLO: Determine the area of a right triangle.
Find the radius when the arc length is 100 units and the given angle is 15 radians
0.15 units
S=rθ; r=θ/S
r= 15/100 = 0.15 or 3/20 units
SLO: Determine the radius of a circle given an angle measure and arc length or sector area.
A 14 ft ladder is leaning against a wall that it is 8 ft away from. What is the angle of depression ? Round to two decimal places.
sin-1(8/14) = 34.85 degrees
SLO:
Solve application problems involving inverse trigonometric functions.
What is the radian measurement and coordinate points for 45o ? What radian measurements and degree does it coordinate with in other quadrants of the unit circle ?
π/4
((√2)/2,(√2)/2)
Quadrant 2: 3π/4 at 135 degrees
Quadrant 3: 5π/4 at 225 degrees
Quadrant 4: 7π/4 at 315 degree
SLO: Determine the exact value of trig functions for all ‘special angles’ (multiples of 30 or 45 degrees) on the Unit Circle.
Write Cot(x) in terms of Csc(x)
1+cot2(x)=Csc2(x)
cot2(x)=Csc2(x)-1
cot(x)= √(csc2(x)-1)
SLO: Make designated trigonometric substitutions and simplify the resulting expressions.
Triangle T and triangle S are similar with corresponding sides in the same proportion solve for x when triangle T’s side A is equal to 37, and side B is equal to 14. While triangle S side A is equal to 8, and side B is equal to X. Round to four decimal places or leave as a fraction.
x= 3.0270 or 112/37
(37/14)=(8/x)
112=37x
SLO: Apply properties of similar triangles to determine their side lengths.
Find the radius of a circle when the given angle is 132 degrees with an area of 145 units. Round to four decimal places
r= 11.2195 units
132 degrees*π/180 = 11π/15 radians
A=(½)θr² ; r=√(2a)/θ)
r= √((2)145)/11π/15) r=11.2195
SLO: Determine the radius of a circle given an angle measure and arc length or sector area.
Determine the exact value of all six trig functions at sinθ=-1/2, in the third quadrant
Sinθ=-1/2
Cosθ=-√(3)/2
Tanθ=1/√(3)
Cscθ=-2
Secθ=-2/√(3)
Cotθ=√3
SLO: Determine the exact value of a trigonometric function given the value of another trigonometric function at the same angle and sufficient information to determine the location of the angle.
What is the radian measurement and coordinate points for 60o ? What radian measurements does it coordinate with in other quadrants of the unit circle ?
π/3
(1/2, (√3)/2)
Quadrant 2: 2π/3 at 120 degrees
Quadrant 3: 4π/3 at 240 degrees
Quadrant 4: 5π/3 at 300 degree
SLO: Determine the exact value of trig functions for all ‘special angles’ (multiples of 30 or 45 degrees) on the Unit Circle.
Does (tan(x)cos(x))/(csc(x)) = cos2(x) ?
If not, what does it equal ?
No
tan(x)cos(x)= (sin(x)/cos(x))*cos(x) = sin(x)
csc(x)= 1/sin(x)
sin(x)/1/sin(x) = sin(x)*sin(x)=sin^2(x)
The function equals sin2(x)
SLO: Verify trigonometric identities by factoring, combining fractions, and multiplying an expression by “1” (a fraction in which the numerator and denominator are identical).
Make designated trigonometric substitutions and simplify the resulting expressions.
List all of the ratios for Sine, Cosine, Tangent and their counterparts
Sin = opposite/hypotenuse ; Csc= hypotenuse/opposite or 1/sin
Cos = adjacent/hypotenuse ; Sec= hypotenuse/adjacent or 1/cos
Tan= opposite/adjacent or tan = sin/cos; Cot=adjacent/opposite or Cot=cos/sin or 1/tan
SLO: List the exact trigonometric ratios for the acute angles of any right triangle.
Find the area of the given circle when the radius is 5 inches and the arc length is 15 in.
37.5 inches2
Arc length; S=rθ
15=5θ; θ=3
Area of a sector of a circle; A=(½)θr²
A=(½)3*(5)²; A= 75/2 inches2 or 37.5 inches2
SLO: Determine the arc length or sector area of a circle given a radius and angle measure.
Determine the exact value of each expression below:
cos-1(sin(5π/6)
sin(arcsin((-√2)/2)
sec-1(2/(√3)
arcsin(cos(7π/a)
tan-1(1/√3)
arccos(sin(5π/6) = arccos(½) = π/3
sin(arcsin((-√2)/2)= sin(-π/4)= (-√2)/2
arcsec(2/(√3)= π/6
arcsin(cos(7π/4)= arcsin((√2)/2)= π/4
tan-1(1/√3)= π/6
SLO:
Determine the exact value of expressions with inverse trigonometric functions at special angles.
Determine the exact value of expressions with compositions of trigonometric and inverse trigonometric functions at any angle.
For the following sine function list the amplitude, phase shift, period, and any vertical shifts.
Where is the midline, maximum and minimum points?
What would the equation of an identical cosine function look like ?
Sketch the graph (graph cannot be included in game, but good practice)
f(x)=4sin(x-2π)+1
Amplitude=4
Phase Shift: right 2π
Period= 2π
Vertical shift: up 1 unit
Midline: y=1
Maximum: (π/2,5)
Minimum: (3π/2,-3)
f(x)=4cos(x-π/2)+1
SLO:
Given the equation of a sine/cosine function:
Determine the amplitude, phase shift, period, and vertical shift of the function
Sketch one complete cycle of the function.
List the points where the function reaches its maximum, minimum, and average value within one complete cycle.
Verify:
(csc2(x)(1-sin2(x))/sec(x) =cos(x)Cot2(x)
(csc2(x)(1-sin2(x)) : 1/sin2(x)*cos2(x) = cos2(x)/sin2(x)
sec(x) = 1/cos(x)
(cos2(x)/sin2(x))/(1/cos(x)) = (cos2(x)/sin2(x))*cos(x) = cos3(x)/sin2(x)
(cos(x)/sin(x))(cos(x)/sin(x)cos(x)
Cos(x)Cot2(x)= cos(x)Cot2(x)
SLO:
Use basic trigonometric identities to simplify trigonometric expressions.
Verify trigonometric identities by factoring, combining fractions, and multiplying an expression by “1” (a fraction in which the numerator and denominator are identical).
Make designated trigonometric substitutions and simplify the resulting expressions.