6. What values of A less than 540° satisfy the equation

7. What are the least eight angles which satisfy the equation

8. Prove geometrically, for all values of A, that

9. Prove geometrically, for angles generally, the formula

sin2 A+ cos2 A = 1.

10. Find all the values of A less than 360° satisfying the equation

2 sin2 A+ sin A = 1.

11. Find all the angles less than 360° satisfying the equation 2 sin2 A = 3 cos A.

12. Find values of less than π satisfying the equation

13. For what angle is the supplement equal to the sine of the complement?

CHAPTER VII.

THE VARIATIONS IN MAGNITUDE AND SIGN OF THE TRIGONOMETRICAL RATIOS.

53. In order that not even the lesser difficulties of the subject may be altogether passed over, the following proposition is more fully stated than those which follow it, which are of a similar nature.

54. To trace the variation in sign and magnitude of sin A, as A increases from 0 to 360°.

Since the symbol representing OP is always positive and does not vary in value, the changes in the value of sin A will all depend upon those of y.

Whilst OP coincides with OA, y is evidently 0;

0

therefore when A = 0, sin A = —; .. sin 0 = 0.

As A increases from 0 to 90°, y increases and is positive; therefore sin A increases and is positive.

When OP arrives at the position OB, PN coincides with OP, and y is therefore equal to r in magnitude; it is positive in sign; therefore

when A = 90°, sin A =

r
r

sin 90° = 1.

As A increases from 90° to 180°, y decreases and is positive; therefore sin A decreases and is positive. When OP arrives at the position OA', PN vanishes; therefore

when A = 180°, sin A = - ... sin 180° = 0.

As A increases from 180° to 270°, y increases and is negative; therefore sin A increases and is negative.

When OP comes to OB', PN again coincides with OP; and y is therefore equal to r in magnitude, but its sign is negative; hence

when A = 270°, sin A = ==; ... sin 270° = −1.

As A increases from 270° to 360°, y decreases and is negative; therefore sin A decreases and is negative. When OP returns to its first position OA, PN again vanishes; therefore

when A = 360°, sin A =

0

sin 360° 0.

=

r

Summary.-Let angle A pass continuously through

all values from 0 to 360°; then

55. To trace the variation in sign and magnitude of cos A, as A increases from 0 to 360°.

With the construction of last proposition, let A be the angle AOP, ≈ the linear units in ON, and r the linear units in OP for all positions of OP. Then

cos A = -.

When A = 0, ON coincides with OP, and
r
cos A= ; ... cos 0=1.

r

As A increases from 0 to 90°, z decreases and is positive; therefore cos A decreases and is positive. When A = 90°, ON vanishes, and

As A increases from 90° to 180°, x increases and is negative; therefore cos A increases and is negative. When A = 180°, ON coincides with OP, and

-r

cos A= ; ... cos 180° = — 1.

As A increases from 180° to 270°, x decreases and is negative; therefore cos A decreases and is negative. When A = 270°, ON vanishes, and

As A increases from 270° to 360°, x increases and is positive; therefore cos A increases and is positive. When A = 360°, ON coincides with OP, and

A tabular summary may be constructed as before (Art. 58).

56. To trace the variation in sign and magnitude of tan A, as A increases from 0 to 360°.

With the same construction as before, let A be angle

F

AOP, y the linear units in PN, and x the linear units in ON for all positions of OP. Then

=

tan A= y

х

When A 0, PN vanishes, and ON coincides with OP; hence

As A increases from 0 to 90°, y increases and a decreases, while both are positive; therefore tan A increases and is positive.

When A = 90°, PN coincides with OP, and ON vanishes; hence

As A increases from 90° to 180°, y decreases and is positive, while a increases and is negative; therefore tan A decreases and is negative.

When A = 180°, PN vanishes, and ON coincides with OP; hence

As A increases from 180° to 270°, y increases and r decreases, while both are negative; therefore tan A increases and is positive.

When A = 270°; PN coincides with OP, and ON vanishes; hence

tan A= ; .. tan 270° = ∞

-r
0

As A increases from 270° to 360°, y decreases and is negative, while x increases and is positive; therefore tan A decreases and is negative.

When A = 360°, PN vanishes, and ON coincides with OP; hence

A tabular summary may, as before, be constructed.