[sp]Outline solution: Write the equation as $2\sin a + 6\cos a = \dfrac{7 - 3\cos b}{\sin b}$. The maximum of the absolute value of the left side is $\sqrt{40}$, occurring when $\cos a = 3\sqrt{40}/20$ (and therefore $\tan a = \pm1/3$). The minimum of the absolute value of the right side is also $\sqrt{40}$, occurring when $\cos b = 1/3$ (and therefore $\tan b = \pm\sqrt{40}/3$). So the equation only holds when both those conditions are satisfied, in other words $\tan^2 a+2\tan^2 b = \dfrac19 + \dfrac{80}9 = 9.$anemone said:Evaluate $\tan^2 a+2\tan^2 b$ if $a$ and $b$ satisfy the trigonometric equality $2\sin a \sin b+3\cos b+6\cos a \sin b=7$.
topsquark said:If I understand what you are doing then that would mean the max and mins of the relevant graphs are the same, thus the LHS and RHS overlap at a point. But the graph shows that that doesn't happen??
-Dan
[sp]That had me fooled too, for a long time. I assumed it meant that there were no solutions to the problem. The explanation (as Mark points out) is that the graph shows two functions of a single variable $x$, but the problem refers to two separate variables $a$ and $b$. So it does not matter that the functions attain their extreme values at different points.[/sp]topsquark said:If I understand what you are doing then that would mean the max and mins of the relevant graphs are the same, thus the LHS and RHS overlap at a point. But the graph shows that that doesn't happen??
-Dan
"tan² a + 2tan² b" is an expression that involves the trigonometric functions tangent (tan) and square (²). The expression is essentially asking you to evaluate the squared tangent of angle a plus two times the squared tangent of angle b.
To evaluate "tan² a + 2tan² b", you will need to know the values of angles a and b. Once you have these values, you can use a calculator or a trigonometric table to find the values of the tangents of these angles. Then, simply substitute these values into the expression and solve for the result.
Yes, "tan² a + 2tan² b" can be simplified. By using the trigonometric identity tan² x = sec² x - 1, we can rewrite the expression as (sec² a - 1) + 2(sec² b - 1). This can be further simplified to 3sec² a + 2sec² b - 3.
Evaluating "tan² a + 2tan² b" can be important in a variety of situations. For example, if you are working on a mathematical or scientific problem that involves angles and trigonometric functions, this expression may need to be evaluated to find a solution. Additionally, understanding how to evaluate this expression can help you better understand and solve more complex mathematical expressions.
One common mistake to avoid when evaluating "tan² a + 2tan² b" is forgetting to convert angles from degrees to radians. Since most calculators and trigonometric tables use radians, it is important to make sure your angles are in the correct units before substituting them into the expression. Additionally, be careful when using the trigonometric identities to simplify the expression, as a simple mistake can lead to an incorrect solution.