Test 4 Review
Some key formulas/pictures to remember:
• Radians to degrees: Multiply π/180◦ .
• 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 and angle with base on x-axis to see this! (“Bowtie diagram” from class)
• Special angles: Recognize them in degrees and in radians. Only focus on Quad I.
• Unit circle coordinates: At angle θ, the point is P (θ) = (x, y) = (cos θ, sin θ).
– Quadrantal angles: Look at the four “poles”.
• ASTC: “All positive” in I, “Sin positive” in II, “Tan positive” in III, “Cos positive” in IV
– Find trig values (by reference angles) and then put on the correct sign.
• Formulas to get reference angles (in radians): When 0 < θ < 2π,
Quadrant
I
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. This explains the range of [−1, 1] and the period 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 more resources, visit my website and look for all the extra downloadable handouts.
WebQuiz 7
#3: “Find the exact values of sin(−9π/2), cos(−9π/2), tan(−9π/2), and cot(−9π/2), whenever possible.
(If there is no solution, enter NO SOLUTION.)”
This is a quadrantal angle. You should add 2π several times to get a value from 0 to 2π. Which point on
the unit circle do you use? Pay close attention to tan and cot; exactly one of them is undefined!
#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(3, 2) and B(5, −8).”
The key ingredient of the terminal side is slope. If you work out the slope of segment AB, then because the
line is parallel, you want the same slope for your side. If that slope is m, the terminal side is y = mx (since
terminal sides always go through the origin).
Pick some convenient x-value in Quadrant II, and then you can get y and r. It may also help to draw a
reference triangle.
#6: “Assume
3 cos(x) − 4 sin(x) =
24
5
4 cos(x) + 3 sin(x) =
7
5
Find the exact (numeric) value of cot(x).”
This is a system of two equations and two unknowns. I recommend the elimination method: for instance, if
you multiply one equation by 3 and the other by 4, then you get coefficients of 12 to cancel each other. Once
you find cos(x) and sin(x), you can get cot(x) = cos(x)/ sin(x).
Bonus question: What quadrant is x in? (Hint: consider the signs of cos(x) and sin(x).)
#7 version 1: “The arc shown is a portion of the
unit circle. Express the AREA of the triangle ∆AOB
as a function of the angle θ.”
#7 version 2: “The arc shown is a portion of the
unit circle. Express the PERIMETER of the shaded
region as a function of the angle θ.”
The point on the unit circle is (cos θ, sin θ). This should tell you a couple sides in each of these figures.
Version 1: You’ll need to find more acute angles. Note that angle ∠AOB must be the complement π/2 − θ,
and thus ∠OAB is its complement θ. Using ∠OAB with SOHCAHTOA, find the base and height you need.
Version 2: One of the sides of the shaded region is the arc length s = 1 · θ. How do you use the cos θ and
sin θ we marked earlier?
WebQuiz 8
Several possibilities for #1 (true/false):
(a) csc(x) = sin
1
x
(b) cot2 (x) − csc2 (x) =
−1
(c) cot(x) = tan( π2 − x)
(d) tan(x − π) = tan(x)
For part (a), what’s the difference between the reciprocal of the function and the reciprocal of the angle?
For (b), start with the Pythagorean identity 1 + cot2 (x) = csc2 (x). For (d), what’s the period of tan(x)?
#2: “Use the information
√
sin(t) =
3
5
π
<t<π
2
to compute cos(t) and tan(t).”
√
Consider a reference version instead: sin(tR ) = 3/5 = opp/hyp. This helps you label two sides of a triangle,
from which you find the third. Which quadrant is t in, and what does that make your signs for cos(t) and tan(t)?
p
#3: “Rewrite csc2 (θ) − 1 in nonradical form without using absolute values for π/2 < θ < π.”
Here, “nonradical” means you’re going to have to get rid of the square root somehow. A Pythagorean
p
identity will help rewrite the root’s body as a perfect square! However, the square root has a subtlety... trig2
is actually |trig| instead of trig. (Basically, your answer must be positive.) Your quadrant info will tell you
whether trig or −trig is positive.
#9: “For the diagram below, you are given that
BD = 2 and CE = 7. Compute the outer perimeter of
the shaded region in terms of θ.”
There are a lot of sides in this problem. It may be
a good idea to first outline a plan on your drawing;
which sides will you find, and in which order? That
will help you organize your thinking.
It’s good to note that there are three right triangles
here: the small triangles ∆ABD and ∆BCE, and the
larger outline ∆ACO. They all have θ as an acute angle (why?). You’ll have to use SOHCAHTOA in more
than one triangle to get the sides you need, though
there is more than one way to finish getting the sides
you need.
It’s also good to notice that opposite sides of the
rectangle have the same length, so DB = OE and
BE = DO.