By Isaac Todhunter
This Elibron Classics ebook is a facsimile reprint of a 1864 version via Macmillan and Co., Cambridge and London.
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Extra info for A Treatise On The Differential Calculus with numerous examples
Thus, we have the following. sin 90Њ = 1 = 1 1 csc 90Њ = 1 1 = 1 sec 90Њ = 1 0 cos 90Њ = 0 = 0 1 tan 90Њ = 1undefined2 cot 90Њ = y A calculator in degree mode returns the correct values for sin 90° and cos 90°. The second screen shows an ERROR message for tan 90°, because 90° is not in the domain of the tangent function. 3 Trigonometric Functions (b) Figure 31 shows the angle. Here, x = - 3, y = 0, and r = 3, so the trigonometric functions have the following values. y θ (–3, 0) 0 25 sin u = 0 = 0 3 csc u = 3 0 x Figure 31 cos u = 1undefined2 sec u = -3 0 = - 1 tan u = =0 3 -3 3 -3 = - 1 cot u = -3 0 1undefined2 Verify that these values can also be found by using the point 1- 1, 02.
However, two similar triangles need not be congruent. (a) (b) (c) Figure 18 The triangular supports for a child’s swing set are congruent (and thus similar) triangles, machine-produced with exactly the same dimensions each time. These supports are just one example of similar triangles. The supports of a long bridge, all the same shape but increasing in size toward the center of the bridge, are examples of similar (but not congruent) figures. See the photo. Suppose a correspondence between two triangles ABC and DEF is set up as shown in Figure 19.
Example 4 Finding Function Values of Quadrantal Angles Find the values of the six trigonometric functions for each angle. (a) an angle of 90° (b) an angle u in standard position with terminal side through 1- 3, 02 Solution (a) Figure 30 shows that the terminal side passes through 10, 12. So x = 0, y = 1, and r = 1. Thus, we have the following. sin 90Њ = 1 = 1 1 csc 90Њ = 1 1 = 1 sec 90Њ = 1 0 cos 90Њ = 0 = 0 1 tan 90Њ = 1undefined2 cot 90Њ = y A calculator in degree mode returns the correct values for sin 90° and cos 90°.
A Treatise On The Differential Calculus with numerous examples by Isaac Todhunter