asked • 2h
First-degree trigonometric equations help
In these questions, find, to the nearest degree, the positive acute angle(s) that satisfy the equation.
1) tan^2x - 4tanx + 3 = 0
2) tan^2x - 2tanx = 3
3) 6sin^2x - 5sinx + 1 = 0
Sindhuja R. answered • 2h
Experienced Tutor specializing in Geometry
1) tan^2x - 4tanx + 3 = 0
A: Let tanx = y
y^2 - 4y + 3 + 0 = 0
By factorization, we get
y^2 - y - 3y + 3 = 0
y (y-1) -3(y-1) = 0
(y-1) (y-3) = 0
y = 1, 3
tanx = 1 => x = 45 degrees
Similarly do the remaining problems, you can ask me anytime for further doubts, i am available at any time
KATHY ZIMMERMANN'S ANSWER:
Welcome. You can learn to do these types of problems quickly with a few simple techniques.
Since equation 1) was done in another tutor's response, and 2) is relatively easy, let's do 3).
Let S = sin of x and S^2 = the square of the sin of x.
(Notice S^2 is not the sin of x^2. That would mean that x was squared before the sin value was obtained. Instead we take the sin of x and then square that value.)
Then the equation is 6 S^2 - 5S + 1 = 0. How do you factor that?
Look at the patterns. We see that 6 = 5 + 1, but that pattern doesn't work when we try (6S - 1) (1S + 1) = 6S^2 + 5S - 1. We need -5S and +1 as shown in the prior paragraph. Even if we changed to -1 in the second factor to get (6S - 1) (1S - 1), it still doesn't give us the correct factoring of 6 S^2 - 5S + 1 = 0. Instead, it produces 6S^2 - 7S + 1. Then using 6 and 1 to get 5 in the middle didn't work.
There are other factors which will give the correct factoring, but we need to know how to find them quickly to avoid going down a long bunny trail that doesn't work on a timed test. Contact me if you want to know this time-saving trick.
The correct factoring turns out to be (3S - 1) (2S - 1), giving 6S^2 - 5S + 1 as we wanted. We set each of the factors equal to zero and solve for S.
3S - 1 = 0, so 3S = 1, and S = 1/3, meaning the Sin of x = 1/3, which is the first sin value.
The next factor is 2S - 1 = 0, 2S = 1, S = 1/2, or Sin x = 1/2, the second sin value.
So how do we convert "sin x = 1/2" into x = the correct angle? One way is to take the inverse sin of (1/2) with a calculator which gives x = 30 degrees. If the answer is in radians, multiply 30 degrees by the factor
(pi radians / 180 degrees) to get pi/6 radians as the angle size.
An alternate technique which will make you speedy and accurate is recognizing common angles and their trig functions. In this case, we can use a 30-60-90 degree triangle. Memorize the proportion of the sides and carry it in your head. If the side opposite the 30-degree angle in a right triangle and is 1, then the hypotenuse is double that (2) and the side adjacent to 30 degrees is 1 * rad 3. It can be proportionate to those numbers: 1: rad 3 :2 or, say, double that: 2: 2 rad 3 :4, etc. Since the sin is found by taking the side opposite (1) over hypotenuse side (2), we have Sin x = 1/2, so 30 degrees is right.
It may seem like a lot of steps, but once you memorize the common angles, you can perform faster on those problems than by using the calculator. The calculator method is to find the Inverse Sin of 1/2. See your manual for how to do it. The inverse Sin of 1/2 = 30 degrees or pi / 6 radians. If you want to know a simple trick to speed up your inverse function for common triangles (like 30-60-90 degrees, or 45-45-90) without a calculator, call me.
Answer 1 was Sin x = 1/3. Take the inverse sin of 1/3 on the calculator to get the angle, .3398 radians, the second possible answer. That's 19.47 degrees.