How to solve without calculator
1. sin(arctan(12/5))
2. cos(arccosx + arcsinx)
Thanks.
You have to know your right triangles.
5 12 13 is one. If the tan is 12/5, then the sin is 5/13. Draw a pic to confirm that.
On the second: Think. Draw any right triangle. Label the angles. Label the hypotenuse as 1, and the one sides as x. The other side will be sqrt (1-x^2). Notice the angle whose sine is x is the complement of the angle whose cosine is x. Does that mean those two angles add to ninety degrees?
can't we do it by taking u=arctan(12/5)
no. arctan(12/5) is not equal to what you started out with.
1.17
To solve without a calculator, you can use your knowledge of right triangles and trigonometric identities.
1. To solve sin(arctan(12/5)), start by considering the right triangle with legs of length 5 and 12, and hypotenuse of length 13 (using the Pythagorean theorem). The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In this case, the tangent of the angle is 12/5. From the triangle, you can see that the sine of the angle is the ratio of the length of the opposite side to the length of the hypotenuse, which is 5/13.
2. To solve cos(arccos(x) + arcsin(x)), you can use the fact that the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, and the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Start by drawing a right triangle and labeling the hypotenuse as 1 and one of the sides as x. The other side will be √(1-x^2) from the Pythagorean theorem. Next, consider the angles whose cosine and sine are x, respectively. Notice that these two angles are complementary, which means their sum is 90 degrees. Therefore, cos(arccos(x) + arcsin(x)) can be simplified to cos(90 degrees), which is 0.
So, the solutions are:
1. sin(arctan(12/5)) = 5/13
2. cos(arccos(x) + arcsin(x)) = 0