For positive acute angles A and B, it is known that cosA= 29/21 and tanB= 40/9 ​ . Find the value of (A+B) in simplest form.

Answer 1

The problem says that cos A = 29/21. But this is impossible, because sine and cosine are always in the range -1 to +1 inclusive. Let's assume that cos A is really 21/29.

cos A = 21/29. Cosine is "adjacent over hypotenuse". So the adjacent side is 21 and the hypotenuse is 29. Using the Pythagorean theorem, we can compute that the remaining side (the opposite one) is 20, because

20² + 21² = 29². So we know all three sides of this triangle!

Tan B = 40/9. Tangent is "opposite over adjacent". So the opposite side is 40 and the adjacent side is 9. Using the Pythagorean theorem, we can compute that the remaining side (the hypotenuse) is 41, because

9² + 40² = 41². So we know all three sides of this triangle!

It's encouraging that every side of every triangle is a whole number.

So we can easily state the value of every trig function of the abgles A and B. For example,

sin A = opposite over hypotenuse = 20/29

sin B = opposite over hypotenuse = 40/41

cos A = adjacent over hypotenuse = 21/29 (they gave us this)

cos B = adjacent over hypotenuse = 9/41

Since we know the above four values, we can use a formula such as

sin (A+B) = (sin A)(cos B) + (cos A)(sin B)

= (20/29)(9/41) + (21/29)(40/41)

= 1020/1189

So the sine of A+B is 1020/1189.

I other words, A+B is the angle whose sine is 1020/1189.

Let's say this in the language of trigonometry:

A+B = arcsin(1020/1189).

Answer 2

To solve for an angle we are going to be utilizing our ArcTrig Functions.

ArcTrig Functions solve for an angle when evaluating (plugging in) a ratio between the side lengths.

Since the problem gives us the cos ratio for Angle A and the tan ratio for Angle B we will be using those ArcTrig Functions to solve the problem.

We want to make sure our calculator is in degrees. ArcTrig Functions look like Cos^-1 and Tan^-1 in a normal scientific calculator.

We would want to plug in the ratios into the ArcTrig function, solve for A and B, and then add them together.

A = Cos^-1 (29/21)

B = Tan^-1 (40/9)

This gives us an error though because the problem itself is stating that the Cos ratio of Adj/Hyp is 29/21. This isn't possible because the adjacent side can not be bigger than the hypotenuse side in a right triangle.

If it's written wrong then the opposite ratio (21/29) is useable to complete the problem, the problem as written is not solvable for that reason.

Here is the problem completely solved written with the 21/29 ratio

A = Cos^-1 (21/29) = 43.60

B = Tan^-1 (40/9) = 77.32

A + B = 43.60 + 77.32 = 120.92

Hope this helps, if you have any further questions please feel free to send me a message!