Download Basic Trigonometry - Answer Explanations

Basic Trigonometry (ACT/PLAN ONLY)
Trigonometry involves ratios between the different sides of a right triangle, relative to a specific
non-right angle. The sine (sin), cosine (cos), and tangent (tan) of a non-right angle are ratios
involving the side opposite (across from) that angle, the side adjacent to (next to) that angle,
and the hypotenuse. These specific ratios are described below and can be remembered by
sounding out the word SohCahToa.
SohCahToa
Sin = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
Below are the three types of problems that make up the somewhat subjectively defined
category of basic trigonometry. For more advanced trig problems, see Advanced Trigonometry.
Problem Type 1: Find the Missing Side
1)
On this type of trig problem, always begin by setting up a trig equation featuring three parts: an
angle and two sides. To figure out which trig function to use, see which two of the three sides
(opposite, adjacent, and hypotenuse) you are dealing with (meaning a side you are given or are
trying to find) in relation to the angle you are given. In this case, you are given the side
opposite the 40˚ angle, and you are trying to find the hypotenuse. Therefore, set up a trig
equation using sine, which is equal to opposite/hypotenuse.
4/sin40˚ 6.22.
sin40˚ = 4/x
xsin40˚ = 4
x=
2)
Begin by setting up a trig equation featuring three parts: an angle and two sides. You are given
the side adjacent to the 20˚ angle, and you are trying to find the side opposite the 20˚ angle, so
you should set up a trig equation in terms of tangent, which is equal to opposite/adjacent.
tan20˚ = x/9 x = 9tan20˚ 3.28.
Problem Type 2: Find the Missing Angle: Inverse Trig Functions
3)
To find an angle using trigonometry, begin by setting up a trig equation much in the same way
as in the above problems. First, consider which two sides you have (opposite, adjacent, and
hypotenuse) relative to the angle you are trying to find. In this case, you have the opposite side
and the adjacent side. Determine which trig function relates those two sides. In this case,
tangent is the appropriate function, since tangent relates the opposite and adjacent sides. Set
up an equation using tangent. tanx = 4/7. To solve, take the inverse tangent (tan-1) of both
sides of the equation. tan-1(tanx) = tan-1(4/7). Because inverse tangent is the inverse function
of tangent, the tan-1 and the tan on the left side of the equation cancel each other out, leaving
you with x = tan-1(4/7) 29.74.
4)
Begin by setting up a trig equation. First, consider which two sides you have (opposite,
adjacent, and hypotenuse) relative to the angle you are trying to find. In this case, you have the
opposite side and the hypotenuse, so cosine is the appropriate function. Set up an equation
using cosine. cos = 3/8. To solve, take the inverse cosine (cos-1) of both sides of the equation.
cos-1(cosx) = cos-1(3/8). Because inverse cosine is the inverse function of cosine, the cos-1 and
the cos on the left side of the equation cancel each other out, leaving you with x = cos -1(3/8)
67.98.
Problem Type 3: Given one Trig Function, Find another Trig Function
5) sinx = 5/13. Find cosx.
Begin by drawing a triangle for which sinx = 5/13. Draw a right triangle and label one of the
non-right angles “x.” Since sin = opposite/hypotenuse, label the side opposite angle x “5” and
the hypotenuse “13.” Your diagram should look something like this. The unknown side has
been labeled a, since it is adjacent to angle x.
Find a by using the Pythagorean Theorem. a2 + 52 = 132 a2 + 25 = 169 a2 = 144 a = 12.
Alternatively, you could have avoided the Pythagorean Theorem if you had recognized that you
had a 5-12-13 Pythagorean Triple. Either way, once you have found that a = 12, it is easy to
figure that cosx = 12/13, since cos = a/h.
6) cosx = 2/5. What is tanx.
Begin by drawing a triangle for which cosx = 2/5. Draw a right triangle and label one of the nonright angles “x.” Since cos = adjacent/hypotenuse, label the side adjacent to angle x “2” and the
hypotenuse “5.” Your diagram should look something like this. The unknown side has been
labeled o, since it is opposite angle x.
Find a by using the Pythagorean Theorem. o2 + 22 = 52
Because tan = o/a, tanx = √ /2.
o2 + 4 = 25
o2 = 21 o = √
.