Elementary Functions and Applications

3. Find the functions of π/8 from those of π/4; of π/12 from those of π/6.

The table given by the Hindu Aryabhata (476 a.d. −) gives the values of the sines of angles at intervals of 3° 45'. How could this table be obtained?

4. If 0 = 90°, show that (1), (2), (3), Section 121, reduce to formulas in Section 62, page 177.

5. If 0 = 180°, show that (1), (2), (3), Section 121, reduce to (1), (2), (3), page 192.

6. State the properties of r3 analogous to each of the equations (1) in Sections 120, 121, 122, 123.

7. In the following, find sin 0 and cos without finding

given

Hint: Find cos 20 from a figure (see Exercise 10, page 170) and then use formulas (1), and (2), Section 123.

8. Express sin 40 in terms of functions of 20; sin 0 in terms of functions of 0/2; sin 30 in terms of functions of 30/2.

9. Transform (3), Section 123, into the forms

Then transform each of the fractions into tan (0/2) by expressing sin and cos in terms of 0/2, using the formulas in Section 122. In what respect is the latter procedure preferable?

10. If tan 20 = 1.4123, find ℗ and cos 0/2.

11. A body is placed on a rough plane which is inclined at any angle greater than the angle of friction. (The angle of friction is an angle such that tan = m, the coefficient of friction.) If the body is supported by a force acting parallel to the plane, find the limits between which the force must lie.

Let be the angle of inclination of the plane, W the weight of the body and R the reaction perpendicular to the plane.

(a) Let the body be on the point of moving down the plane, so that the force of friction acts up the plane and is equal to mR. Let P be the force required to keep the body at rest.

Resolving W into components parallel and perpendicular to the plane, we have P+mR = W sin 0, R = W cos 0, and therefore P W (sin - m cos 0). Since m = tan we have, P W (sin

tan & cos 0).

(b) Let the body be on the point of motion up the plane. Complete the

solution.

12. What is the maximum and the minimum force which will hold a weight of 12 pounds on a plane inclined at an angle of 40°, if the coefficient of friction is 0.5, and if the force acts parallel to the plane?

13. A block W rests on a horizontal plane. If an oblique force P acts upon W, making an angle with the direction of sliding, and if the coefficient of friction is m = tan, prove that the magnitude of P that will cause the block to slide is

Find the least pull that will make the block slide.

Suggestion: Show that P will be a minimum when 0 = 4, i.e., the direction of pull is given by the angle of friction.

14. If the inclination of a plane is 0 = arc sin, what horizontal force would support a body of 50 pounds upon it? What horizontal force would be required if 0 were doubled?

15. On the same axes, plot the graphs of the functions below. Find the period of each function.

(e) sin a cos (ẞ − a) + cos a sin (ẞ − a) = sin B.

124. Sum and Difference of the Sines or Cosines of Two Angles. To express the sum of the sines of two angles, sin a + sin ẞ, as a product, let

Then sin a + sin ẞ = sin( 0 + $) + sin (0 – 6)

Solving the first two equations for ✪ and ø, by adding and subtracting, and dividing by 2, we have

0 = (a + B), =
Ꮎ 1 $ 1/2 (a - ẞ),

and hence, substituting,

sin a + sin ẞ = 2 sin (a + ß) cos (a – ß).

In like manner, prove

(1)

– B).

(2)

(3)

(4)

sin a sin ẞ = 2 cos(a + B) sin B cos a + cos ẞ = 2 cos (a + B) cos (a – ẞ). cos a cos B = − 2 sin (a + B) sin (a – ẞ). Equations (1) and (2) give properties of the function sin x. What are the analogous properties of log x? To what property of 22 is (2) analogous?

125. Logarithmic Solution of Triangles, Case III. The law of sines (page 201) may be written