ACT Math: Advanced Trigonometry Problems

One of our students has been working hard on the Math section of the ACT.  He’s a very good student, so he’s been focusing a lot of his attention on the last ten problems where many of the more difficult topics are tested.  Every ACT Math test contains four questions that are classified as “Trigonometry.”  These questions can test the trigonometric relationship in right triangles, the properties and graphs of the trigonometric functions, and trig identities and equations.  These questions go beyond the basic SOH-CAH-TOA problems that we discussed in a previous post.

Right triangle trig is usually introduced to students in Geometry, but the really rigorous examination of the topic comes later on in Pre-Calculus.  There we learn that angles can be measured in something called radians as well as in degrees.  You can convert an angle from degrees to radians by multiplying by π/180.  The equivalent degree and radian measures for a number of important angles are shown in the diagram below.

Source: Henrico County VA Public Schools
Source: Henrico County VA Public Schools

We also learn in Pre-Calculus that the values of sin θ and cos θ can be negative depending upon where the angle θ is located on the unit circle.  To learn more about coordinates on the unit circle, read this helpful article from Purdue University-Calumet.

Consider this example:

If cos θ = -15/17 and π < θ < 3π/2, what is the value of sin θ ?
(A) -8/17
(B) -8/15
(C)  8/15
(D) 8/17
(E) 17/8

The first thing you want to do is sketch the triangle, paying attention to where it is located on the xy-coordinate system and what the sides of the triangle will be, both in terms of their values and their signs.

Advanced Trig Example 1

You’ll notice that since the problem told us that the angle θ is between π and 3π/2, we’ve drawn our triangle in the third quadrant and that both the horizontal and vertical sides of our triangle are labeled with negative values.  As you label the triangle, use the fact that SOHCAHTOA tells us that cosine is the ratio of the adjacent side to the hypotenuse.  The remaining side can be found using the Pythagorean Theorem (or you may remember that 8-15-17 is a “famous” right triangle, one of the ones it’s helpful to know going into the test).

Now let’s answer the question.  We were asked to find sin θ, and we know from our knowledge of SOHCAHTOA that sine is the ratio of the opposite side to the hypotenuse. From our picture, we can see that sin θ = -8/17, so the correct answer is Choice (A).

Another area of trigonometry that is normally not taught until students take Pre-Calculus is the trigonometric identities.  There are a large number of these but perhaps the most important one, and the one most likely to appear on the ACT, is this:  sin2  θ + cos2  θ  = 1.

Now let’s have you try a few of these advanced trigonometry problems.  The answers appear at the bottom of this post.

1)    If sin θ  = 12/13 and π/2 < θ < π, then tan θ = ?

(A)  -12/5
(B)  -12/13
(C)  5/12
(D)  5/13
(E)  12/5

2)    If tan θ = 3/4 and  π < θ < 3π/2, then cos θ = ?

(F)  -4/3
(G)  -4/5
(H)  -3/5
(J)   3/5
(K)  4/5

3)    If the value, to the nearest thousandth, of sin θ = -0.259, which of the following could be true about the value of θ?

(A)  0 < θ < π/6
(B)  π/6 < θ < π/2
(C)  π < θ < 7π/6
(D)  7π/6 < θ < 3π/2
(E)  3π/2 < θ < 11π/6

4)    If the value, to the nearest thousandth, of cos θ  = 0.309, which of the following could be true about the value of θ?

(F)  0 < θ < π/4
(G)  3π/4 < θ < π
(H)  5π/4 < θ < 3π/2
(J)   3π/2 < θ < 7π/4
(K)  7π/4 < θ < 2π

5)    For all values of θ, (2sin θ  – 2cos θ )2 = ?

(A)  4sin2 θ – 4sin θ cos θ  + 4cos2 θ
(B)  4sin2  θ + 4cos2 θ
(C)  -4sin θ cos θ

(D)  -8sin θ cos θ
(E)  4 – 8sin θ cos θ

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at info@cardinalec.com .

Solutions:
1)  (A)
2)  (G)
3)  (C)
4)  (J)
5)  (E)

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