Is sec Θ ° > tan Θ ° for every angle?
Please explain why or why not?
Find a positive number x for which cos (ln x) = 0.
I really don't know where to start, please help me.
let's look at their graphs
http://www.wolframalpha.com/input/?i=y+%3D+1%2Fcosx+%2C+y+%3D+tanx
for some values, the secant curve is above the tangent curve, for others the opposite
easiest way is to look at the CAST rule
in quad IV , the secant is positive, but the tangent is negative, so secØ > tanØ
in quad III, the secant is negative, but the tangent is positive, so secØ < tanØ
e.g. Ø = 315°
sec 315° = √2
tan45 = -1
sec45 > tan45 in IV
Ø = 225°
sec225 = -√2 = -1.414..
tan225 = 1
thus tanØ > secØ in III
you second question....
If cos(lnx) = 0
then lnx could be π/2 , since we know cos(π/2) = 0
so if ln(x) = π/2
then x = appr 4.8105
check:
set your calculator to radians
take ln(4.8105)
now press 'cos' to get close to zero
may be easier to recognize that
sec = 1/cos
tan = sin/cos
sec > tan implies
1/cos > sin/cos
which is true iff
cos > 0
that's in quadrants 1 and 4
Thank you both, reiny and jolly rancher.
Thanks for the graph, and jolly rancher, that totally made it a lot easier for me to understand. :)
To determine whether sec Θ ° is greater than tan Θ ° for every angle, we need to understand what secant and tangent represent.
In trigonometry, secant (sec) is the reciprocal of cosine (cos), while tangent (tan) is the ratio of sine (sin) to cosine (cos).
To compare sec Θ ° and tan Θ °, we can rewrite them using their respective definitions:
sec Θ ° = 1 / cos Θ °
tan Θ ° = sin Θ ° / cos Θ °
Let's examine the relationship between these two trigonometric functions.
For any angle, cos Θ ° can be zero or positive. However, it cannot be negative because it represents the ratio of the adjacent side to the hypotenuse in the unit circle, and both sides are always positive.
Now, if cos Θ ° is zero, sec Θ ° will be undefined since division by zero is not possible. Therefore, sec Θ ° is not defined for angles where cos Θ ° = 0.
On the other hand, for angles where cos Θ ° is positive, sec Θ ° can be greater than, equal to, or less than tan Θ °, depending on the values of sin Θ °.
By comparing the definitions of sec Θ ° and tan Θ °, we can see that sec Θ ° will be greater than tan Θ ° when sin Θ ° is less than 1 but not equal to zero. In this case, the reciprocal of a positive number (cos) will be greater than the ratio of two numbers (sin/cos).
However, when sin Θ ° is equal to zero, then tan Θ ° will be equal to zero, and sec Θ ° will be undefined (since division by zero is not possible).
In summary, sec Θ ° is not greater than tan Θ ° for every angle. It depends on the values of sin Θ ° and cos Θ °.