Precalc Trigonometric Units (units 10, 11, 12)
Simplify tan x/csc x - sin x/cos x
(sin x - 1)/cot x
Simplify the expression fully. sin u + cot u * cos u
(sin^2u + cos^2u)/sin U = csc u simplified by eliminating cot
Simplify (cot (-theta) + tan (-theta))/(cot (theta))
- sec^2 (theta)
Simplify 1/(cos(π/2-θ) - 3 (csc θ)
-2 csc θ
Evaluate without a calculator: sec 37° - csc 53°
0 (zero)
What is the process for determining the general form solutions to trigonometric equations
1. graph the left hand side and right hand side of the equations 2. look for the intersecting points 3. identify repeating patterns - solutions to the right of maximum are considered downhill - solutions to the left of the maximum are considered up hill 4. write the general form of the equation
What is the final simplified form of the following expression in terms of sine and/or cosine? sec u * cot u
1/sin u
You have an angle of depression of 54° from the top of the ramp into the water when the water is at a level of 671.54 feet. How much ramp is needed to get you to floating docks? The entire depth of lake travis is 681 feet.
11.69 feet
To convert radians to degrees multiply by __________
180/pi
At the UTHS office building, a swinging pendulum sits inside of a cylinder with diameter of 10 meters and a height of 150 feet. The pendulum is designed as tall as possible to fit inside of the cylinder. The pendulum motor at the top can be set to what maximum swing angle (as measured at the top)?
3.8 degrees
Find all solutions to cos x = -squareroot (3)/2 in the interval 0<=x<=2pi
5pi/6 and 7pi/6
A jet ski leaves a beach restaurant and travels 3 miles out to sea. The jet ski is spotted from a coast guard, who is 4.85 miles away from the beach restaurant on land. The jet ski has an angle of 46 to both the restaurant and the coast guard. How far is the jet ski from the coast guard?
6.68 miles
What is the equation for calculating the area of a triangle
A = 1/2 * b*c* sin (α) or A = 1/2 * a *c* sin(β) where α is the angle opposite side A and β is the angle opposite side B need to be given 2 sides with included angle
How do you calculate the area of a segment of a circle
As = Area of Sector - Area of Triangle As = 1/2* r^2*θ - 1/2*r^2*sin(α) θ must be in radians to use this equation
What is the domain and range for arcsin(x)
Domain [-1, 1) Range [-pi/2, pi/2] see graph
What are the domain and range for arccos(x)
Domain [-1, 1] Range [0, pi] see graph
What are the domain and range for arctan(x)
Domain [-infinity, infinity] Range [-pi/2, pi/2] see graph
Write the Law of Sines
Sin A/a = Sin B/b = Sin C/c
How do you calculate the area of a triangle when you have 3 sides (SSS)
Use Heron's Formula A =squareroot(s(s-a)(s-b)(s-c)) s=1/2(a+b+c)
Is the following a valid identity sin x + cos (x) * cot (x) = csc (x)
Yes show by graphing shows they lie on top of each other or algebraically converting left side of equation to csc (x)
Angle of Elevation- When an observer views an object above eye level, an angle of elevation is formed between ______________
a horizontal line at the observer's eye level and the line connecting the observer to the object. (object is above horizontal line)
What is the Law of Cosines
a^2 = b^2 + c^2 - 2bc Cos a or the other 2 forms on your formula sheet
The general form solution to trigonometric equations finds __________
all solutions not just solutions over a given interval
How is a bearing angle measured
clockwise rotation from due North ex. 80° (must distinguish from standard angle 80°)
Which trigonometric functions are even?
cos (x) = cos (-x) sec (x) = sec (-x)
If f(x) = cos (x) then f^-1(x) = ____
cos^-1(x)
How is a standard angle measured
counterclockwise rotation from the x axis ex. 80° (must distinguish from bearing angle 80°)
Definition of even identity is? What is the symmetry of even identity?
f(x) = f(-x), symmetric about the y axis
f^-1(x) is the standard notation for an _____________ of a given function f(x)
inverse
How is a directional bearing angle measured
number of degrees east or west of North or South ex N40°E or S70°W
what are the Pythagorean Identities?
see formula sheet
Simplify csc u - cos u * cot u
sin u
If f(x) = sin (x) then f^-1(x) = _____
sin^-1(x)
In alternating current circuits, current oscillates through the circuit in a back and forth motion. The voltage (in Volts), therefore, changes with time in regular intervals and can be defined in terms of time by the function below. V(t) = 120 sin(120pi *t) where t is the time in seconds. At what times does the voltage equal -120 volts? Round your answer to four decimal places.
t = 0.0125 + 0.0167 *n it hits -120 volts many times since it is an alternating circuit
If f(x) = tan (x) then f^-1 (x) =
tan^-1(x)
Simplifying Trig Expressions: one term is better than ______ two trig functions is better than _____ and clearly, one trig function is better than ________!
two terms three trig functions two
When is the Law of Sines used to solve trigonometric problems
when you do not have a right triangle and 2 angles and 1 side are given or 1 side and 2 angles are given
When do you use Law of Cosines
when you do not have a right triangle and you are given a SAS or SSS
Find all general form solutions to sin(-x) = squareroot(3)/2
x = 4pi/2 + 2n*pi and x = 5pi/3 + 2n*pi
Solve cos x = - squareroot(3)/2 using inverse trig to find one solution.
x = 5pi/6
Find all general form solutions to sin x = 1/2
x = pi/2 + 2n*pi and x = 5pi/6 + 2n*pi
Solve 5-5tan(x) = 0 using inverse trig to find one solution.
x = pi/4
What is the general form solution for 2 cos x = squareroot (2)
x = pi/4 + 2pi *n and 7pi/4 + 2pi *n where n is any integer
When can you use Law of Sines without checking for ambiguity
If you are given two angles and a side (AAS or ASA)
Cofunction identities are 2 functions shifted on x axis by _______
pi/2
What is the process for identifying solutions to trigonometric expressions by graphing
1. Enter the left hand side and right hand side of the equations in your calculator in different rows and graph each 2. Look for the intersection points on for the given interval
Is the following a valid identity tan^2(x)/(sec (x) - 1) = csc (x)
No show by either graphing or using data tables
To convert degrees to radians multiply by ________
pi/180
Find all solutions to sin x = squareroot(2)/2 on the interval 0<=x<= 2pi
pi/4 and 3pi/4 9pi/4 and 11pi/4 but not in the given interval
Find all solutions to sin x = 1/2 in the interval [-pi/2 to pi/2]
pi/6 -7pi/6 and 5pi/6 are solutions but not in the given interval
Which trigonometric functions are odd?
sin (x) = - sin (x) csc (x) = - csc (x) tan (x) = - tan (x) cot (x) = - cot (x)
In right triangles complementary angles add to 90 and the sin θ = cos(_________) and cos θ = sin ( ____________)
sinθ = cos (90-θ) and cosθ = sin(90-θ)
Solve cos (x) = 1/2 using inverse trig to find one solution.
x = pi/3
Find the general form for all the solutions to the equation below. tan^2 (x) = 3
x = pi/3 + n*pi and x = 2pi/3 + n*pi
What are the forms for inverse cosine
θ = arccos(x) or θ = cos^-1(x) or x = cos(θ)
What are the forms for inverse tangent
θ = arctan(x) or θ = tan^-1(x) or x = tan(θ)
A wheel chair ramp needs to be redesigned because the current angle is too steep, resulting in people having difficulty getting their wheelchairs up the ramp during rainy or snowy weather. You are asked to redesign a new ramp. The current ramp is made from a 10 foot long piece of wood and makes an angle of elevation of 15 degrees with the ground. How long of a ramp is needed so that the new angle of elevation would be decreased to 10 degrees? Round your answer to the nearest tenth of a foot.
14.9 ft
A solar street light is 34 ft above the curb of a street, and lights all the way to the middle of the street. If the street is 230 feet wide, determine the angle of elevation from the middle of the street to the top of the street light.
16.47 degrees
A surveyor can measure the height of very tall objects (such as mountains) by making two measurements using the following procedure: Step 1: From ground level, stand such that you can view the peak of the mountain and measure the angle of elevation. Step 2: Walk directly toward the mountain, record the distance you walked and measure the angle of elevation at this new spot. Step 3: Sketch out and label an obtuse triangle such that the vertices are the two locations where you took measurements and the third vertex is the peak of the mountain. Step 4: Use the given information to label the triangle as ASA. Then use the Law of Sines and right triangle trig to determine the height of the mountain. A surveyor measures the angle of elevation to the peak of Mt. Everest as 45 degrees. She then walks towards mountain for 1 mile and then measures the new angle of elevation as 50.7 degrees. What is the height of Mt. Everest (in miles) based on these calculations? Round to the nearest tenth of a mile.
5.5 miles
What does it mean to check for ambiguity when you use Law of Sines
If you are given 2 sides and an angle (SSA or ASS) You must check for the possibility of no triangle, one triangle, or 2 triangles exist from the given information (check for ambiguity)
Angle of Depression- When an observer views an object below eye level, an angle of depression is formed between
a horizontal line at the observer's eye level and the line connecting the observer to the object. (object is below horizontal lin)
Definition of odd identify is?
f(x) = -f(-x), symmetric about the origin
While trig functions take an angle as the input and produce a ratio (between -1 and 1) as an output, inverse trig functions take _____________
ratios and produce angles as outputs.
What are the cofunction identities for sin, cos, tan, cot, sec, csc
see formula sheet
What are the quotient identities for tan and cot
see formula sheet
What are the sum and difference identities for sin and cos
sin (a+b) = sin a cos b - cos a sin b sin (a-b) = sin a cos b - cos a sin b cos (a+b) = cos a cos b - sin a sin b cos (a-b) = cos a cos b + sin a sin b
Find all general form solutions to 2cos x = 0
x = pi/2 + n*pi *Here in the equation, "x" represents an angle, not the x-coordinate. If this may be causing confusion, just get in the habit of rewriting arguments as theta instead of x. Writing cos(\theta) = 0 may help you avoid confusion when using the unit circle. Since we know "cosine" represents the adjacent side of the triangle formed by \theta, therefore we can use the x-coordinate.
Solve 2sin(x) = 1 using inverse trig to find one solution.
x = pi/6
What are the forms for inverse sine
θ = arcsin(x) or θ = sin^-1(x) or x = sin (θ)
What is the process for identifying solutions to trigonometric expressions with the unit circle
1. Highlight the interval of the expression on the unit circle 2. Look for the point on the circle that gives the answer sin x = squareroot(3)/2 x = pi/3 is the angle of the unit circle where sin x = squareroot(3)/2
Equation is an identity if both sides of the equation are ___________
equal for All input values
Find all solutions to tan^2 ( x ) = 1 in the interval [0, 2pi)
pi/4, 3pi/4, 5pi/4, and 7pi/4 There should be four solutions. If you found only two of them, you may have forgotten that with equations of degree two, there is a negative and positive version...in this case, where tangent equals plus or minus one.
Express sin(50°) in three different ways using the sum and difference identities
sin (60-10) = sin 60 * cos 10 - cos 60 * sin 10 sin (30 + 20) = sin 30 * cos 20 + cos 30 * sin 20 sin (40 + 10) = sin 40 * cos 10 + cos 40 * sin 10
What are the reciprocal identities for sin, cos, tan, csc, sec, cot
see formula sheet
What are the 3 ways to check and see if an identity is valid
1. algebraically, 2. graphing the left hand side and right hand side to see if they lie on top of each other, 3. use a data table to see if the values match
Draw a right triangle formed by perpendicular segments AB and BC. Then, solve for b if a = 7 and m∠C = 27°.
13.74 Always draw a visual. he question is asking for the length of segment AC, which could be labeled as lower case b since it should be drawn opposite from angle capital B. Angles and their opposite sides are labeled with the same letter. The angles are labeled with capital letters and side lengths are labeled with lower case letters.
A missile is sent on a bearing of 083. After some time it changes bearing to 125. Sketch the path as two bearings (head to tail) and calculate the measure of the angle between them. Write your answer as degrees rounded to the nearest hundredth.
138 degrees
