Sections 5.2A (cont) and 5.2B: Uses of Right Triangles, and Trig on the Unit Circle
Two Equations, Two Unknowns Revisited in a Two-Layer Situation
You’ll have some questions giving you two equations sort of 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). This is really another “two-layer” problem!
1. The outer layer of the problem is finding two unknowns in two equations. Let’s abbreviate the unknowns,
say C = cos x and S = sin x. We get
C + 2S = 11/5
2C + S = 10/5
which can be solved by the elimination method.
2. Once you know C and S (i.e. cos x and sin x), now you use them to get any other trig function you want.
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 acute angles whose trig values can be computed by hand, called the special angles. They
are 30◦ , 45◦ , and 60◦ .1 (In radians, they are π/6, π/4, and π/3.) You should memorize these values for both
degrees and radians:
θ in degrees θ in radians sin θ √
cos θ √
tan θ
30◦
π/6
1/2
3/2
3/3
√
√
◦
2/2 √1
45
π/4
√2/2
60◦
π/3
3/2 1/2
3
(Remember that once you know sin and cos, you can get all the trigs.)
NOTE: To remember this, look at the numerators. For sines, the tops go
is reversed! In fact, cos(60◦ ) = sin(30◦ ) and vice versa.
√
1,
√
2,
√
3. For cosines, the order
Ex 2: If a right triangle has a 30◦ angle, and its adjacent leg is 6, find the lengths of the other two sides.
Basic Applications of Right Triangle Trig
When you have a situation with a given angle and a right triangle, and you want to know sides, we use trig.
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.
TO START: To find the adjacent leg, use cos = adj/hyp. For the opposite leg, sin = opp/hyp.
NOTE: Some problems say that you can type “deg” for degrees, as in “sin(35deg)”. Otherwise, you would
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.
TO START: Note the angle of 39◦ is made with the ground. That’s how you know which angle to label.
Ex 6: If AD = 5 in the figure ∆ABE, find some “link” between
on the right, find AB and BC.
the top and middle triangles: the
common hypotenuse of AE! Now
APPROACH: You know a side you have a side and angle of ∆ABE.
and angle in the “top” triangle
∆ADE, so you can get any sides
Similarly, you’ll want to find BE
you want of it. To get AB, though, to get a “link” between the triangles
which is in the “middle” triangle ∆ABE and ∆BCE.
1 30◦ and 60◦ come from splitting up an equilateral triangle, and 45◦ comes from an isosceles triangle. If you want to know more
about where these angles come from, talk to me outside of class.
Trig Functions of Arbitrary Angles
SOHCAHTOA only works for acute angles. We have a different definition for arbitrary angles.
The Trig Functions of Any 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.
NOTE: In Quadrant I, draw a right triangle where opp = y, adj = x, and hyp = r (so θ is the angle made
with the x-axis). You get the same formulas as SOHCAHTOA!
Identities: Every identity we learned in the last class still works with this definition! For instance, we still
have tan = sin / cos and sin2 + cos2 = 1, etc. However, the trig functions aren’t always positive anymore!
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. NOTE: θ is a right angle!
Some problems tell you the terminal side of the angle by giving you a line and a quadrant. Pick P (x, y) to
be any point on the line in the correct quadrant; choose x yourself with the correct sign, then find y.
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.
TO START: Get the slope of the terminal side: m = −1/(−9) = 1/9. Next, terminal sides MUST go through
the origin, so the terminal side 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 solely on the quadrant of the angle. This is because it depends on the
signs of x and y (which is determined by the quadrant).
Quad I: All trigs are positive.
Quad II: Sin is positive (as is its
reciprocal csc). The others are negative.
Quad III: Tan is positive (as is
cot).
Quad IV: Cos is positive (as is
sec).
Handy figure:
S
T
A
C
“A Smart Trig Class”
To find an angle’s quadrant, we usually rewrite it as a coterminal value from 0◦ to 360◦ (or from 0 to 2π).
However, multiples of 90◦ (i.e. of π/2) are quadrantal and do not belong to a quadrant!
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
Quadrantal angles: If θ is quadrantal, then the point P (x, y) is on an axis: we usually use (±1, 0) or (0, ±1).
Two of the six ratios use division 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.