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Math
Geometry
Trigonometry
TRIG EXAM 1
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Terms in this set (44)
one complete revolution equals?
one degree equals
360 degrees
- 1/360 degrees of a full revolution
what are the degrees, specifically surrounding 90 degrees, of an acute angle?
- an angle that measures less than 90 degrees
what are the degrees, specifically surrounding 90 degrees, of an right angle?
an angle that measures 90 degrees or a 90 degree angle.
what are the degrees, specifically surrounding 90 degrees, of an obtuse angle?
- an angle greater than 90 degrees
what are the degrees, specifically surrounding 90 degrees, of a straight angle?
an angle that measures 180 degrees
If two angles have a sum of 90 degrees, what are they called?
- complementary angles
90 degrees - alpha
If two angles have a sum of 180 degrees, what are they called?
- supplementary angles
180 degrees - alpha
How does the equation look when it states to give the complementary and supplement of each angle?
90 - y
180 - y
what does a little line (tick) mean if its going through a line of one or more sides of a triangle
- the ticks can mean all sides are the time (if every side has one tick) or all sides are different (if every side has a different amount of ticks).
- the ticks indicate which sides are different in length
where are the leg b, a, c (hypotenuse), and theta sign located on a right traingle?
- hypotenuse (c) is located on the longest side of the right triangle, A is located on the bottom side and B is located on the left side. Theta is located on the inside of the triangle at the pointer end.
what is the equation for the pythagorean theorem?
a^2 + b^2 = c^2 Remember c is the hypotenuse.
What are the degrees for a special triangle that has the same measurements on all sides? Where do the degrees belong on a special triangle? what is the name for this triangle?
30 degrees - 60 degrees - 90 degrees. Yes they do, all measurements are the same. The 90 degrees in one corner of the triangle (bottom), the 30 degrees belongs on the top corner and the 60 degrees belongs on the right hand corner.
- A special triangle is called an Equilateral Triangle (all sides are the same length).
What is the ratio of a 30-60-90 triangle? (special triangle)
t : square root 3 times t : 2 x t (aka 2t)
what are the two types of special triangles (in degrees)
30-60-90
45-45-90
what is the ratio for a 45-45-90 degree triangle?
t : t : square root 2 times t
what is the vertex of a square?
- the common point (the pointy side of the triangle)
where are the first, second third, and fourth quadrants located on a graph. In each quadrants, what the are signs (positive and negative) for x and y?
- first quadrant (top right) x and y coordinates are positive.
- second quadrant (top left) x is negative and y is positive
- third quadrant (bottom left) x is negative and y is negative)
- fourth quadrant (bottom right) x is positive and y is negative
What is the distance formula?
square root of (x2-x1) squared + (y2-y1) squared
when finding the equation of a circle what letters signify the center point and what letter signifies the radius?
(h,k)
r
we use the distance formula involving the radius (r) and the center points (h,k) which replace (x1 and y1)
Trigonometric Functions
sin θ = y/r
tan θ = x/r
sec θ = r/x
cos θ = x/r
cot θ = x/y
csc θ = r/y
what formula do we use to find the remaining trigonomic functions?
r squared = x squared +y squared (Pathegreom Theorm)
what are the reciprocal identities
1. cscθ = 1/sinθ
2. secθ = 1/cosθ
3. cotθ = 1/tanθ
sin = 1/cosθ
cos = 1/secθ
what are the ratio identities?
cotθ x/y = cosθ/sinθ
tanθ = sinθ / cosθ
what is the rule to memoize the signs of trigonometric functions in a given quadrant?
CAST rule.
- starting from the bottom right and going counter clockwise, we write one letting in each Q. C A S T
C : stands for cos
A: stands for All
S: stands for sin
T: stands for tan
First Q it says "All" meaning sin, cos, and tan are positive--we use only sin, cos, and tan because those are the originals
Second Q it says Sin which means Sin is positive and the other two (tan and cos) are negative
Third Q it says Tan which means Tan is positive and the other two are negative.
Fourth Q it says Cos which means Cos is positive and the other two are negative.
if sinθ = (professor inputs number) and cosθ = (professor puts in number), find the exact value of each of the four remaining trig. functions. What do you do?
- first you would plug in the numbers given by professor of sinθ and cosθ.
- next, use reciprocal identities to solve the rest of the trig functions
Pythagorean Identities
cos²θ +sin²θ=1 -- equivalent to / sin²θ = 1 - cos²θ / cos²θ = 1 - sin²θ
1+tan²θ=sec²θ
cot²θ+1=csc²θ
when simplifying an expression as much as possible using ratio and reciprocal identities and you run into a problem where you need to take the square root of of a trigonometric function (tan^2θ) how will the answer look?
tanθ^2 will have to be positive if we are taking the square root so tanθ^2 will be between absolute value bars and look like this |tanθ|
when working on transforming the left side of the equation into the right side, what is the main goal?
- to make the answer of the left side of the equation the same as the right.
- we work on the LHS first and leave the RHS alone
- our goal is to have the LHS and RHS have the same answer.
Right triangle definition:
Sin θ =
Cos θ =
Tan θ =
Cot θ =
Sec θ =
Csc θ =
Sin θ - opposite / hypotenuse
Cos θ - Adjacent / Hypotenuse
Tan θ - Opposite / Adjacent
Cot θ - Adjacent / Opposite
Sec θ Hypotenuse / Adjacent
Csc θ Hypotenuse / Opposite
Right triangle definition:
sin A =
cos A =
tan A =
cot A =
sec A =
csc A =
sin A = a/c
cos A = b/c
tan A = a/b
cot A = b/a
sec A = c/d
csc A = c/a
where is hypotenuse, adjacent, and opposite place on a right triangle?
hyp - placed on the longest side
adjacent - places under the triangle
opposite - place on the side of the triangle
(draw one for a visual)
How do you find sin B ?
- use a right triangle that is labeled with hypotenuse side, opposite side, and adjacent side and use Sin A answers for guidance.
if two angles have a sum of 90 degree, what are they called?
- they are called complementary angles
what is the cofunction theorem?
- a trigonometric function of an angle is always equal to the cofunction of the complement of the angle
sin A = cos B
tan A = cot B
sec A = csc B
Use calculator to find Trig. Functions and Angles:
Solve: tan 45.29
what do we do?^
we need a calc to solve
- first plug in the number 45.29
- second, press "tan" button
- next, round number to nearest given degree (optional)
Use calculator to find Trig. Functions and Angles: Use Reciprocal Identities to Evaluate. Use sin -1, cos -1 and tan -1 to find theta
secθ 18.7
cotθ 52.52
what do we do?
first- we need to use reciprocal identities to change secθ to 1/cosθ
second - use to the 1/x button to find the reciprocal of the number
third - plug into calculator:
18.7
cos
divide symbol
1/x button
do not forget to write answer in appropriate significant figures if question does not specify to round
same process for cotθ
first - use reciprocal identities to change cotθ to 1/tanθ
Find the acute angle θ. Round your answer to the nearest tenth of a degree
sinθ 0.7139
cscθ 1.8214
first - type of number given 0.7139
second - press 2nd button
third - press sin button
first - 1.8214
second - 1/x
third - 2nd
fourth - sin
sin B
tan B
sec B
cos B
cot B
csc B
sin B - b/c
tan B - b/a
sec B - c/a
cos B - a/c
cot B - a/b
csc B - c/b
what is a equilateral triangle?
Triangles are equilateral when all three sides have the same length
what is an isosceles triangle
- two sides have the same length
What is a scalene triangle?
all three sides of the triangle have a different length
what is an acute triangle?
a triangle with three acute angles (special triangle)
obtuse triangle
- the sum of three interior angles is 180 degrees
what is a right triangle
A triangle containing a 90° angle.
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