How to do the following Trigo. Question?

Question

tan 1/2 x = -1/2

I'd like an explanation of how to do it instead of just the answer please! Btw, for those who don't want to dig out a calculator, I understand. inverse tan 1/2 =26.6 degrees

Answer 1

First of all, it would help to clarify the given problem a bit, to eliminate any ambiguity:

tan [(1/2)x] = - (1/2)

Let's begin by substituting an easier-to-write variable 'u' in place of tangent's argument '(1/2)x'.

tan u = -1/2

The most important thing for you to realize is how the tangent function is made:

"Tangent takes the ratio of two legs of a right triangle, namely the opposite side over the adjacent side."

The right-hand side of the equation above, then, means "the ratio of opposite to adjacent is -1 / 2."

Can you recall a 'special triangle' (30-60-90 or 45-45-90) whose legs are in this same ratio? No. So, we're forced to rely on a calculator to tell us the precise angle whose tangent is -1/2. As you pointed out, we may use the inverse tangent to do this:

u = InvTan (-1/2)
u = -26.57

Before going further, let's remember that 'u' stood for '1/2 x', so we must now solve for x:

1/2 x = -26.57

x = -53.13

Notice that x, being negative, is an angle drawn in Quadrant IV of the Cartesian plane. Draw a right triangle as follows:

o The adjacent leg lies along the x-axis
o The opposite leg is parallel to the y-axis
o The hypotenuse radiates from the origin, torward 'southeast' or 'the lower right-hand direction' or 'about 4:30 on a clock face'; the precise angle at the origin is x = -53.13.

Your original statement:

tan 1/2 x = -1/2

simply states that "if you drew a triangle like the first one, but the angle at the origin was HALF the original angle, you'd end up with a triangle whose opposite and adjacent sides were in the ratio 1:2." That's right, 1:2 or 1/2. The only additional comment to make, since the given fraction -1/2 is negative, is that one of the legs must be oriented 'downward' or 'leftward'. Is that the case with our drawn triangle? Yes! The opposite (vertical) leg indeed points down, into Quadrant IV.

As an exercise, you might draw right triangles in all four quadrants and ponder how their adjacent and opposite legs 'go' and what effect this has on their trigonometric functions (sine, cosine, tangent). Many teachers instruct you to memorize "ASTC" or "All Students Take Calculus" so that you remember which trig function is positive in each quadrant:

Quadrant I = All
Quadrant II = Sine
Quadrant III = Tangent
Quadrant IV = Cosine

Knowing such things gives you a big head start in trig class!

Hope this helps!

Answer 2

Answer: x = -53.2º

Explanation: To understand intuitively any trigonometrical question, it helps a lot to think that at the end we are talking about angles. They are angles and then they are the tangent, the sinus or the cosine of these angles. So, in our case, when we talk about "tan 1/2 x = -1/2", we are talking about "the tangent of a certaiin angle, that is in this case 1/2 x, being equal to -1/2"

Let's re-write the problem as follows: tan (angle) = -1/2
(where, of course, angle = 1/2 x, but this is not important in a first stage).

The idea to solve the problem is then: we'll first search for what angle tan(angle) = -1/2. Later, as a second step, we'll find x, based on the fact that angle = 1/2x.

STEP 1) So, let's go with the first step. solve the equation tan(angle) = -1/2. They are two ways to solve trigonometrical equations:

1) Based on known facts, like that tan(45º) = 1, and so if tan(angle) = 1 we can conclude that angle = 45º.

2) Based on approximations, when the values are not known. For example, if tan(angle) = -1/2 we cannot conclude directly what is the angle, because there's no a well-known value for it. Instead of this, we rely on the fact that tan is a continuous and monothonically growing function when angle goes from -90º to 90º, and so we search by approximations an angle between -90º and +90º such that tan(angle) = -1/2 (this is the general procedure to define any irrational number).

Fortunately the calculator does this job for us, and in our case the answer is tan(-26.6º) = -1/2 (as you mentionned). There is no other method to find the value. You must go with approximations, manually or via calculator. The value is close to -26.6º (more exactly, close to -26.57º), that is all we can say (we can continue adding and adding decimal numbers to our approximation, and in fact the calculator gives around 10 or 20 more, but to have an exact value we should continue to infinite values, so just writing -26.6º is enough)

So, we found: angle = -26.6º (aprox).

STEP 2) Now, remember that angle = 1/2 x, so we have that x = 2*angle, and so x = 2*(-26,6º) = -53.2º (aprox).

We found the answer: x = -53.2º (aprox.) is the solution to the equation tan 1/2 x = -1/2

Answer 3

Fine. That inverse tan 1/2 = 26.6 degrees gives you the related angle (relevant angle just might be the better word choice, but talking trigo... related has the slight edge)

That means the .5x argument is related to 26.6 degrees. Which in turn means x has related angle of 53.2 ...but since principal values for inverse tan come out of the range -90 to +90 and your tangent function is negative

you must choose the related angle to be in quadrant IV:
x = -53.2

Answer 4

because it truly is a good triangle, attitude CAB is 40 5 degrees. First, commence by having the length of Line ad. considering all of us do not ignore that attitude CAB is 40 5 degrees, attitude DAB must be 40 5-20=25 degrees. To get the length of Line ad, use: COS=adjoining/Hypotenuse, it truly is COS 25 degrees= 29/Line ad. COS 25=.906. We get .906AD=29; divide out the .906 and also you get a length for Line ad as 29/.906= 32 cm. Now that we've the length of line ad, we are able to ascertain the length of line CD by creating use of: Tan=opposite/adjoining. Tan 20 degrees=Line CD/32. We proceed to get .364=Line CD/32. Multiply both aspect by 32 to verify the length of Line CD and it measures .364*32= 11.sixty 5 cm.

Answer 5

Tangent is negative in second quadrant.[Tangent=opposite side/adjacent side. Note that opposite side is positive but adjacent side is negative in second quadrant] The angle has to be 90+26.6=116.6 deg

Answer 6

Read your book...