Test 2

Test 2

tan^-1(1.92) is the angle whose tangent is 1.92.

Call this angle theta; theta tan^-1(1.92). We don't need to find theta to find the sine of this angle.

This angle theta would be contained in any number of right triangles of different sizes.

One of these triangles would have angle theta opposite a side of 1.92 and adjacent to a side is 1.

The hypotenuse of this triangle is sqrt(1.92^2 + 1^2).

The sine of the angle would therefore be

sin(theta) = opposite side / hypotenuse = 1.92 / sqrt( 1.92^2 + 1^2).

Note that all our information on this triangle comes from our knowledge of the tangent.

We conclude that sin(tan^-1(1.92) ) = 1.92 / sqrt( 1.92^2 + 1^2).

Class notes contain pictures of similar situations. In general, if you have a situation involving the inverse sine, inverse cosine, inverse tangent, inverse secant, etc. of a number, you can draw a right triangle with two known sides (one equal to the number, the other equal to 1 as in this example; which sides are which depend on which inverse function you are looking at) and use the pythagorean theorem to find the third. From this you can find the cosine, sine, etc. without ever finding the angle.

Test 2

Test 2

tan^-1(1.92) is the angle whose tangent is 1.92.

Call this angle theta; theta tan^-1(1.92). We don't need to find theta to find the sine of this angle.

This angle theta would be contained in any number of right triangles of different sizes.

One of these triangles would have angle theta opposite a side of 1.92 and adjacent to a side is 1.

The hypotenuse of this triangle is sqrt(1.92^2 + 1^2).

The sine of the angle would therefore be

sin(theta) = opposite side / hypotenuse = 1.92 / sqrt( 1.92^2 + 1^2).

Note that all our information on this triangle comes from our knowledge of the tangent.

We conclude that sin(tan^-1(1.92) ) = 1.92 / sqrt( 1.92^2 + 1^2).

Class notes contain pictures of similar situations. In general, if you have a situation involving the inverse sine, inverse cosine, inverse tangent, inverse secant, etc. of a number, you can draw a right triangle with two known sides (one equal to the number, the other equal to 1 as in this example; which sides are which depend on which inverse function you are looking at) and use the pythagorean theorem to find the third. From this you can find the cosine, sine, etc. without ever finding the angle.