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Sections 5.2A (cont) and 5.2B: Uses of Right Triangles, and Trig on the Unit Circle
Two Equations, Two Unknowns Revisited with Trig Unknowns
You’ll have some questions giving you two equations with trig functions, like
cos x + 2 sin x =
11/5
2 cos x + sin x =
10/5
and after that, you need to find trig functions like sec(x) or cot(x).
1. Solve for the two unknowns cos x and sin x in the two equations. You may wish to abbreviate these
unknowns with new variables1 , like A = cos x and B = sin x, to get A + 2B = 11/5 and 2A + B = 10/5.
I recommend using the elimination method to solve this.
2. Once you know cos x and sin x, now you use them to get any other trig function you want. Remember
any trig can be written in terms of sines and cosines!
Ex 1: Assume cos(x) + 3 sin(x) = 15/5 and 2 cos(x) + sin(x) = 10/5. Find sec(x) and tan(x).
Special Angles
There are a few angles, called special angles, where we can easily find the trig values by hand. They are
30◦ , 45◦ , and 60◦ .2 You should memorize these sine and cosine values for both degrees and radians...
tangent values are handy but less important. (And recall tan = sin / cos anyway.)
θ in degrees
30◦
45◦
60◦
θ in radians
π/6
π/4
π/3
sin θ
√1/2
√2/2
3/2
cos θ
√
√3/2
2/2
1/2
tan θ
√
3/3
√1
3
√ √ √
Mnemonic device: Look at the numerators: 1, 2, 3. For sine, the values grow as the angle grows. For
cosine, the values decrease instead, so it looks like the reverse of the sine column.
Ex 2: If a right triangle has a 30◦ angle, and its adjacent leg is 6, find the other sides without a calculator.
Basic Applications of Right Triangle Trig
Use SOHCAHTOA when you have a right triangle, you know an acute angle, and you want to find sides.
1. Find out which two sides interest you (one is known, the other is the unknown).
2. Write down their trig ratio with those sides (usually sin, cos, or tan). Solve for the unknown.
Ex 3: A right triangle has a hypotenuse of 3 and an acute angle of 35 degrees. Determine the area and
perimeter of the triangle.
Getting started : hyp is known. Use cos = adj/hyp to find adj. Use sin = opp/hyp to find opp.
Notation: Some problems let you type “deg” for degrees, like “sin(35deg)”. Otherwise, you have to convert
to radians: “sin(35π/180)”.
Ex 4: The base of a right circular cone has diameter 12 inches. The angle between the radius and the
slanted edge is 45 degrees. Determine the height and volume of the cone.
Ex 5: A radio transmission tower is 33 feet tall and makes a right angle with the ground. A guy wire
(supporting wire) is to be attached to the tower 20 feet from the top of the tower and makes an angle of 39
degrees with the ground. Determine the length of the guy wire.
Hint: The angle of 39◦ is made with the ground, not with the tower.
Ex 6: If AD = 5 in the
figure on the right, find
AB and BC.
Approach: We could
find all parts of the “top”
triangle ∆ADE, since we
know an angle and side for
it.
AB is in the “middle” triangle ∆ABE, so
we should find something
the triangles have in common: the common hypotenuse AE! Compute
AE, and now you have a
side and angle of ∆ABE.
Similarly, you’ll need a
side in common between
the “middle” ∆ABE and
“bottom” ∆BCE, which
means you’ll compute the
leg BE as well.
Finally, you know a side of
∆BCE...
1 Think back to the “u = ax ” trick we used in Section 4.6. You don’t have to substitute the unknown with a new variable name,
but it can help make the problem look less cluttered, and it lets you focus on solving one kind of problem at a time.
2 30◦ and 60◦ come from splitting up an equilateral triangle, and 45◦ comes from an isosceles triangle.
Trig Functions of Arbitrary Angles (SOHCAHTOA is only for acute angles)
Trig Functions for ANY real θ: Draw θ in standard
position. Let (x, y) be a non-origin point on its terminal side, and let r be its radius (distance to the origin),
so that x2 + y 2 = r2 . Then we have
sin(θ) =
cos(θ) =
tan(θ) =
y
r
x
r
y
x
csc(θ) =
sec(θ) =
cot(θ) =
r
y
r
x
x
y
In particular, note x = r cos(θ) and y = r sin(θ) . On the unit circle, x = cos θ and y = sin θ.
To remember this: Draw a vertical leg to make a right triangle. If you call opp = y, adj = x, hyp = r, you
get the same formulas as SOHCAHTOA! (However, “opp” and “adj” can be negative or zero now.)
Identities: The identities we learned still work with this new definition! For instance, tan = sin / cos (when
cos 6= 0) and sin2 + cos2 = 1, etc. However, don’t assume the trig functions are positive now !
Ex 7:
(a) If P (24, −7) is on the terminal side of θ, find sin(θ), cos(θ), and tan(θ).
(b) Do the same steps if P (0, 4) is on the terminal side. Hint: θ is a right angle!
Some problems tell you the terminal side of the angle by giving you a line and a quadrant. We get a point
P (x, y) on the line in that quadrant by picking any x with the correct sign, then finding y from the line.
Ex 8: Find the exact values of sec(θ) and csc(θ) if θ is in standard position and the terminal side of θ is in
Quadrant IV and on the line 2x + 7y = 0.
Ex 9: Find the values of sin(θ) and cot(θ) if θ is in Quad III and its terminal side is perpendicular to
y = −9x + 4.
Getting started : The slope of the terminal side is m = −1/(−9) = 1/9. The terminal side must go through
the origin (why?), so its equation is y = (1/9)x. Repeat the steps of Ex 8... I’d use x = −9 (why?).
Signs of the Trigs
The sign of a trig function depends only on the quadrant, i.e. on the signs of x and y. In every quadrant
except Quad I, only one of sin, cos, or tan is positive: the other two are negative.
Quad I: All trigs are positive.
Handy figure:
Quad II: Sin is positive.
S
T
Quad III: Tan is positive.
Quad IV: Cos is positive.
A
C
Warning: This doesn’t apply to
the multiples of 90◦ (i.e. of π/2)!
Use a point on the circle instead.
(See “quadrantal angles” below.)
“A Smart Trig Class”
Finding quadrant: Get your angle from 0◦ to 360◦ (or from 0 to 2π) by using a coterminal value. (Add or
subtract multiples of 360◦ or 2π.)
Ex 10: For each angle, find its quadrant.
(a) 500◦
(b) −5π/3
(c) 17π/2
Ex 11: Find θ’s quadrant if...
(a) cos(θ) < 0 and sin(θ) > 0
(b) sec(θ) > 0 and tan(θ) < 0
Hint for (b): sec is 1/ cos, so it has the same sign as cos.
Quadrantal angles: The point P is on an axis: we usually use (±1, 0) (North and South points) or (0, ±1)
(East and West points). Two of the six ratios divide by zero, so exactly two trig ratios are undefined.
Ex 12: For each of these quadrantal angles, find all six values of the trig functions. (Say “NO SOLUTION”
for any undefined values.)
(a) −450◦
(b) 17π
Look online for another handout. We’ll also continue with signs next class.