Download Tuesday Mar 25 - UGA Math Department

Test 4 Review
Some key formulas/pictures to remember:
• Circle formulas: s = rθ, A = 12 θr2 with θ in radians
• SOHCAHTOA for right triangles: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
– The other three trigs are reciprocals, e.g. sec = 1/ cos. Also, tan = sin / cos.
– For arbitrary points (x, y) with radius r: replace opp with y, adj with x, and hyp with r
– Draw a reference triangle with base on x-axis to see this! The reference angle also touches x-axis.
– If you’re given a trig value, then use its positive version to make a reference triangle.
• Special angles: Recognize them in degrees and in radians. Only focus on Quad I.
• Unit circle coordinates: At angle θ, P = (cos θ, sin θ).
– For quadrantal angles, you draw this circle and look at its four “poles”.
– For quadrantals: Every function except sin and cos is either 0 or undefined. For sin and cos, they
could also be ±1.
• ASTC: “All positive” in I, “Sin positive” in II, “Tan positive” in III, “Cos positive” in IV
– Usually, we find trig values (by reference angles) and THEN put on the correct sign.
– If no quadrant or sign info is given, use both signs.
• Formulas to get reference angles (in radians): These require 0 ≤ θ < 2π.
I
Quadrant
To get θR : θR = θ
Solve these for θ: θ = θR
II
θR = π − θ
θ = π − θR
III
θR = θ − π
θ = θR + π
IV
θR = 2π − θ
θ = 2π − θR
You should also be able to understand these by drawing a picture, not just by memorizing this table.
• Look back over trig identities from Sections 5.2A and 5.3... they occur in several different notes pages.
• Look at the graphs from last week... you should know what the domains, ranges, and periods are.
Summaries to Keep in Mind:
• Coterminal angles: you can add or subtract 2π’s (aka 360◦ ’s) without changing trig values.
– When solving equations, don’t forget about periodicity: add and subtract 2π’s as appropriate.
• sin and cos are the easiest trigs to use:
– They are the only ones which are defined everywhere.
– All other trigs can be written in terms of sin and cos.
– They are the coordinates of the UNIT circle. In fact, this shows why their range is [−1, 1] and their
period is 2π.
• When you draw right triangles in a figure:
– If you know two sides, get the third using Pythagorean Theorem.
– If you know a side and an acute angle, use SOHCAHTOA to get more sides.
• Almost any trig value problem can be first attacked with a reference angle and a 1st-quadrant picture.
Find the ratio first, then put the right sign on!
Selected WebQuiz 7 and 8 Problems with Hints
For an overview of more problems, visit my website and look for an extra downloadable handout. Don’t
forget to also look for my extra attachments for this unit! WQ 7 #6 is covered on one of those.
WebQuiz 7
#2: “Find the EXACT values of the indicated trigonometric functions. Decimal approximations will be
marked incorrect. (a) sec(30◦ ) (b) cot(60◦ )”
For this, write sec as 1/ cos and cot as either 1/ tan or cos / sin. You don’t have to rationalize your answers!
#3: “Find the exact values of the six trigonometric functions of 13π/2, whenever possible. Your answer
should be an exact value with no trigonometric functions. (If there is no solution, enter NO SOLUTION.)”
This is a quadrantal angle (multiple of 90◦ or of π/2). It’s not between 0 and 2π, so subtract 2π a couple
times to get it that way: you get π/2. What point on the unit circle is that? Once you know the coordinates
(cos is x-coord, sin is y-coord), you can get the six trigs.
#5: “Find the exact values of the trigonometric functions of θ if θ is in standard position and the terminal
side of θ is in Quadrant II and is parallel to the line through A(5, −8) and B(6, −11).”
First off, do you remember what “standard position” means? If not, review 5.1’s notes.
The key ingredient of the terminal side is slope... compute it from AB! Because the line is parallel, you
want the same slope (not the opposite reciprocal). Now, the terminal side ALWAYS goes through the origin.
Once you have a convenient point, you can either draw a reference triangle, or you can find x, y, r. See for
yourself why both approaches would produce the same result.
#6 modified: “If x is an acute angle, express sin(x) in terms of sec(x).”
This kind of problem is covered in an extra handout. The basic idea is to label a right triangle using
SOHCAHTOA and the equation sec(x) = sec(x)
= hyp
1
adj . You can figure out opp/adj from that.
#7: “Assume 3 cos(x) − 4 sin(x) = 4 and 4 cos(x) + 3 sin(x) = 3. Find the exact value of cot(x).”
This is a system of equations with cos(x) and sin(x) as the two unknowns... abbreviate them as A and B
if you wish. I’d use the elimination method twice: set up one elimination to get A, and set up another to get
B. (Use substitution if you want, but I found the common-denominator work more annoying that way in this
problem.) Once you know sin(x) and cos(x), you can get cot(x).
#8 modified: “A sector has area 9, radius r and angle x in radians. Express r and the perimeter of the
sector as functions of x.”
This is really short. The sector area is given; write that area formula, and you can get r on its own. (You’ll
have to cancel r2 , so your last step will be a square root.) Once you know r, write down the three sides of the
sector and add them to get perimeter.
WebQuiz 8
Possibilities for #1 (true/false):
(a) sec(x) = cos
1
x
(b) tan2 (x) − sec2 (x) =
−1
(c) tan(x) = cot( π2 − x)
(d) cot(x − π) = cot(x)
(a) is wrong; you want the reciprocal of the function, not the angle! For (b), start with the Pythagorean
identity 1 + tan2 (x) = sec2 (x). For (c), it’s a complementary identity. For (d), tan and cot both have period π.
#2: “Find the RADIAN measure of the √
angle in the interval [0, π/2] which makes the trigonometric function
2 3
value correct. (a) cos(α) = 1 (b) csc(β) = 3 ”
Since the angle is in [0, π/2], only the first quadrant is used, PLUS the two quadrantal points bordering it
(the right and top poles of the unit circle). Use special angle values and quadrantal values! For (b), it may help
to simplify the reciprocal by dividing the powers of 3.
p
#4: “Rewrite the expression csc2 (θ) − 1 in nonradical form without using absolute values for 5π/2 < θ <
3π.”
For this, “nonradical” means you’re going to have to get rid of the square root somehow... it turns out that
2
if you use an appropriate
√ identity, the expression underneath the root can be written as one perfect square a !
2
However, recall that a is not always a... it’s |a|. In other words, the square root symbol must give a positive
answer. To know whether you should use a or −a, use the given info to get a quadrant and hence the correct
sign.
#7: “A square in inscribed in a circle of radius 15
as shown. Determine the area of the shaded region.”
It’s easiest to subtract the square’s area from the
circle’s area. The approaches involve breaking it up
into right triangles by drawing lines. My favorite approach is to use both horizontal and vertical lines to
break the square into four right triangles, since every
leg length is the circle radius!
(You don’t actually need trig to do this problem,
but we’ll see a variant in Unit 5 that DOES need trig.)
#10: “The arc shown in the figure is a portion of
the unit circle, x2 + y 2 = 1. Express the PERIMETER
of the shaded region as a function of the angle θ (a
radian measurement).”
You can get the dimensions of the triangle by using the coordinates of the point on the unit circle:
(cos θ, sin θ). You can then find all three sides of the
shaded region; note that one of them uses arc length!