Help with finding the exact value?
Since you're given tan(a) = -4/3, and tan = opp/adj, make a right angle triangle with angle a and opposite side equal to 4 and adjacent side equal to 3. By the Pythagorean theorem, the hypotenuse should be equal to sqrt (3^2 + 4^2) = sqrt (25) = 5. For that reason, it follows that sin(a) = opp/hyp = 4/5 (it is positive because in quadrant 2, sine is positive). Similarly, cos(a) = adj/hyp = -3/5. (It is negative since "a" is given to be in quadrant 2, where cosine is negative). Since cos(b) = 2/3, and cos = adj/hyp, make a right angle triangle with adj = 2 and hyp = 3. It then follows that opp = sqrt(3^2 - 2^2) = sqrt(9 - 4) = sqrt(5). Therefore, sin(b) = opp/hyp = sqrt(5)/3 By the cos addition identity, cos(a + b) = cosacosb - sinasinb And since we know that cos(a) = -3/5 sin(a) = 4/5 cos(b) = 2/3 sin(b) = sqrt(5)/3 cos(a + b) = (-3/5)(2/3) - (4/5) (sqrt(5)/3) cos(a + b) = (-2/5) - (4sqrt(5)/15) cos(a + b) = (-6 - 4sqrt(5)]/15
I need help in finding exact values and also some other math questions?
I'm really confused. I have no idea how to do the exact value for cos 13pi/6. The first thing i did was find the degrees and it came out to 390 degrees then I did 390-360 and got 30. I'm confused what to do now. The next problem is the same thing but it's sin 14pi/3. I'm totally stuck on this one. The next one is suppose 0< theta
How do I find the exact value of [math] \sin 1.5 [/math]?
As one of the previous post mentioned, sin(1.5) is irrational so the exact value of it is in fact sin(1.5). But I think we can do better than that, since we are using trig functions that are not intuitively easy to grasp.Lets first start by representing sin(1.5) as an integral:So essentially what we are doing when we are evaluating sin(1.5) is evaluating the area under the curve of [math] 1.5*\int_{0}^{1} cos(1.5x)dx[/math]We can graph this intergral to get a better understanding of the value of sin(1.5).Here is a graph of the Integral that I made on desmos:Now it becomes more clear that sin(1.5) is just under the half peroid cycle of the function [math]1.5cos(1.5x),[/math] with the half peroid going from [math][0,pi/3]. [/math]We can also represent this as a Maclaurin series:[math]1.5*\int_{0}^{1} cos(1.5x)dx = 1.5*(\sum\limits_{n=0}^\infty\frac{(-1^n)(1.5x)^{2n}}{(2n)!})[/math]So while finding the exact value is impossible, here are some solution that make it more intuitive to what the answer is.I hope this helps!
(math help please?) Find the exact values by using sum or difference identies.?
(1) 5π/12 can be written as 2π/12 + 3π/12 = π/6 + π/4 cos(5π/12) = cos(π/6 + π/4) = cos(π/6)cos(π/4) - sin(π/6)sin(π/4) Exact values. cos(π/4) = sin(π/4) = 1/√2 and cos(π/6) = √3 / 2 sin(π/6) = 1/2 Putting these values in cos(π/6)cos(π/4) - sin(π/6)sin(π/4) = (√3 / 2 )(1/√2) - (1/2)(1/√2)) = √3 / 2√2 - 1 / 2√2 = (√3 - 1) / 2√2 We can multiply top and botom by √2 [(√3 - 1) / 2√2] X [√2 / √2] = √2(√3 - 1) / 2√2.√2 √2((√3 - 1) / 4 Answer C (2) cos5°cos40° - sin5° sin40° = cos( 5 + 40)° = cos45° = 1/√2 = (1/√2) X (√2 /√2 ) = √2 / √2.√2 = √2 / 2 Answer D Note exact values : cos45° = sin45° = 1/√2.
Find the Exact Value of cos(285°)?
Have you the formula cos (A+B) = cos A cos B - sin A sin B? Then cos (285°) = cos (75°) .... [4th quadrant, cos positive, 360° - 285°] = cos(45°+30°) = cos 45° cos 30° - sin 45° sin 30° = (1/√2)(√3/2) - )1/√2)(1/2) = (√3 - 1)/(2√2) = (√6 - √2)/4 Do cos 15° the same way, but use (45° - 30°). You should get this answer with +√2 replacing the -√2. By the way, if you're interested in geometry, there's a nice way of doing this by extending the (2, 2, 2) equilateral triangle with altitude √3 from which we get the ratios for 30° and 60°. Extend the altitude beyond the vertex for a distance of 2 units, making it √3 + 2 altogether. Join the top of it to one of the other vertices of the triangle, and then you have an isosceles triangle formed by the two 2 unit sides, with angles 15°, 15°, and 150°. Its longest side is the hypotenuse of a rightangled triangle whose other sides are √3 + 2 and 1. Applying Pythagoras' finds that its length is √(8 + 2√12) = √ 6 + √2, and then we can read off the trig ratios of 15° and 75°, using surd operations to express them all with rational denominators.
How do I find the exact value of [math]\cos\left(\tan^{-1}\left (\frac{15}{8}\right)+\tan^{-1}\left (\frac{4}{3}\right)\right)[/math]?
Before we go about answering this question, let us think for a moment.What we have is cosine of “sum of two angles”.The first angle is such that its tangent is 15/8. I can make a guess that this angle should be a little more than 60 degrees. How is that possible? It is because the tangent of 60 degrees is equal to [math]\sqrt{3}[/math], which is approximately equal to 1.7, whereas 15/8 is between 1.8 and 1.9.The second angle is such that its tangent is 4/3, which is approximately 1.3. Therefore, that angle should be between 45 degrees and 60 degrees.Hence, the sum of angles is between 105 degrees and 120 degrees. This gives us two hints about the answer: (1) The magnitude of the answer is much closer to zero than it is to one. (2) The answer is in fact negative. Therefore, the answer should be between -0.5 and zero, and hence, the answer suggested by the OP in the question details is definitely wrong.The most straightforward way to calculate the answer is to use:cos(A+B) = cos(A)cos(B)-sin(A)sin(B)where A = arctan(15/8) and B = arctan(4/3)All the quantities on the right hand side can be computed by constructing right triangles. For the angle A, the hypotenuse is 17, and for the angle B, the hypotenuse is 5.Hence, we have sin(A) = 15/17, cos(A) 8/17, sin(B) = 4/5, cos (B) = 3/5.Plugin, and solve, you will get an answer between -0.5 and 0.
If [math] \sinh x = \frac{3}{4} [/math], find the exact value of [math] \tanh x [/math]?
Thanks for A2A.Hint: Use [math] \cosh^2 x - \sinh^2 x =1[/math], [math]\tanh x = \frac{\sinh x}{ \cosh x}[/math] and [math]\cosh x > 0[/math] [math]\forall x \in \mathbf{R}[/math].
Math problem,how do you find the exact value for sin17π/6?
change improper fraction into proper fraction: sin(17pi / 6) = sin(5pi/6 + 2pi) 2pi is 360 degres which is one revolution. Now draw a terminal arm extending from the origin outwards in a cartesian plane which makes an angle of 17pi/6 with the x-axis. When you draw it the angle it makes with the x - axis is the same as 5pi/2. This is because the angle 17pi/6 is just 5pi/6 with an extra revolution, so you end up in the same place. So the sine value for 17pi/6 is the same as for 5pi/6 (you can ignore any extra revolutions!). So: = sin(5pi/6) now since 5pi/6 is not an acute angle (5pi/6 > pi/2 = 3pi/6), write it in terms of acute angles and multiples of pi: = sin[(6pi - pi) / 6] = sin[pi - pi/6] since (pi - pi/6) is in the second quadrant sine value is positive. = sin(pi/6) = 1/2 some helpful formulas: quadrant 1 (all are positive): no formulas. These are just normal trig values. quadrant 2 (only sin positive): sin(pi - theta) = sin(theta) cos(pi - theta) = -cos(theta) tan(pi - theta) = -tan(theta) has to be in (pi - theta) format. quadrant 3 (only tan is positive): sin(pi + theta) = -sin(theta) cos(pi + theta) = -cos(theta) tan(pi + theta) = tan(theta) has to be in (pi + theta) format. quadrant 4 (only cos is positive): sin(-theta) = -sin(theta) cos(-theta) = cos(theta) tan(-theta) = -tan(theta) has to be in (-theta) format. also: sin[(2pi) * n + theta] = sin(theta) where 'n' is any integer, basically (2pi) * n = multiples of 360 degrees. It works with any other trig functions too. you can also use formula for two angles: sin(alpha + beta) = sin(alpha)cos(beta) + sin(beta)cos(alpha) cos(alpha + beta) = cos(alpha)cos(beta) - sin(alpha)sin(beta) tan(alpha + beta) = [tan(alpha) + tan(beta)] / [1 - tan(alpha)tan(beta)]