What is the angle of inclination of the straight line that passes through the point (1,4) (6,0)?

Question

I got the slope (-4/5) from the two points above. I plugged the slope I used the angle of inclination therom as so tan x = -4/5 and then arranged and plugged it in my calculator as so tan-1(-4/5) = -0.674740942 rad

however, my textbook says its 2.47 rad

any ideas as to what I did wrong?

Answer 1

You did nothing wrong except you didn't finish.

The angle of inclination is defined as the positive angle that the line makes with the horizontal.

Calculators will always give the 4th quadrant angle for a negative tangent (and for a negative sine as well). You need to know and, more importantly, understand, that there is a 2nd quadrant positive angle answers well.

So you need add pi radians to your 4th quadrant calculator

answer to get the correct answer to this problem. (When working in degrees you add 180 deg.)

It is kind of like when you take a square root. The calculator can only give one answer so it gives the positive and you need to know and understand that there is also a negative. Try it. Enter -5 then square it and get 25. Then hit the square root key and get 5, not -5.

Back to this problem. Your calculator is obviously in radian mode when you got -0.674740942. Now add pi using the pi key and you will have 2.466851711. Now take the tangent and you will see that it is -0.8 exactly.

(You might want to always do this as a check anyway.)

Answer 2

Sofia

Your method wasn't wrong but the angle of inclination is not defined in the way you assume. I just Googled the definition and the angle of inclination of a line is the angle the line makes with the POSITIVE x-axis (so not the reference angle necessarily). So the angle of inclination is the angle in radians measured counter clockwise from the x-axis to the line. (And the angle of inclination needs to be positive using that definition), so if you think about your line you will see the angle needs to be pi minus 0.67 which gives the textbook answer. So your textbook answer is correct following the definition.

Mike

Answer 3

Hi Sophia,

When you find the inverse tangent on your calculator, your calculator returns an angle that is between -pi/2 and pi/2, that is, the angle is located in the first or fourth quadrant.

The book is giving you an angle that lies in the 2nd quadrant. If you take your -0.675 and simply add pi to it, you get 2.47 (which lies in the 2nd quadrant).

Hope this helps!

JMJ

Answer 4

Come on Sofia P.,

Add those two radians values together (as absolute values). You should notice something familiar in the sum! You are not "wrong", and neither is your textbook. They represent the same slope. Unless, by convention, you are limited to a particular set of range values?

-- Cheers, --Mr. d.

P.S. In the future, try graphing each solution, when you have a discrepancy. That should have "caught" your assumption of a difference. And also, if you solve an algebraic equation to get the value of an unknown (or more than one!), always check each solution vs. the original problem. This is particularly an issue with solving equations in which one of the many solutions has a quadratic factor in it. Some, or even all, of the algebraic solutions may be impossible (the swimming pool had -10 feet of water in it; the cliff was -150 m high, the trip took -1.2 hours and consumed -4 gallons of fuel, etc.) in real life. Especially don't put all these scenarios together (drive -1.2 hours, consuming -4 gallons of fuel, then plunge over a -150 m cliff into -10 feet of water). Ick! you might end up with -1.2 of a car. just sayin'.

-- Cheers, --Mr. d.