If the revolution be continuous, the values of the sine are periodic, every successive revolution indicating a new cycle, and a new wave in the curve. The sines are equal for pairs of angles symmetric about the y-axis. OTHER TRIGONOMETRIC CURVES. The tangent is 0 for the angle 0; increases through the first quarter to ∞; leaps to∞; increases through the second quarter to 0; increases through the third quarter to; leaps to; increases through the fourth quarter to 0; and so on. It has all values from -∞o to +∞o. The tangents are equal for pairs of angles that differ by a half-revolution. The secant is equal to the radius and its ratio is +1 for the angle 0; increases through the first quarter to +∞; leaps to ∞; increases through the second quarter to the opposite of the radius and its ratio is -1; decreases through the third quarter to; leaps to t∞; decreases through the fourth quarter to the value at beginning; and so on. It has no value smaller than the radius. The secants are equal for pairs of angles symmetric about the x-axis. The cosine, cotangent, cosecant have the same bounds as the sine, tangent, secant; they go through like changes and are represented by like curves; but they begin, for the angle 0, with different values, viz. : the radius, ∞, ∞. EXAMPLES. 1. Show directly from the definitions what are the largest and what the smallest values that each function may have, and state for what angles the several functions take these values. 2. Draw the curve of tangents, curve of secants, curve of cosines, curve of cotangents, and curve of cosecants. 3. Trace the changes, when ✪ increases from 0 to 2 π, in: § 4. FUNCTIONS OF RELATED ANGLES. In the theorems that follow, the reader may examine the different figures and apply the proofs to each of them; he will find no angle in any quarter to which they do not apply, and thus he will see that they are general. In particular, he may note the qualities of the lines for angles in the different quarters. For, let xop be any plane angle 0, and take Q symmetric with and 0, the distances of P, Q are equal, their abscissas identical, and their ordinates opposite; i.e., •.• r'=r, x' = x, y' = −y, always, [geom. FUNCTIONS OF THE SUPPLEMENT OF AN ANGLE. THEOR. 6. If be any plane angle, then: sin π-0=+sin 0, cosπ· ·0=-cos 0, tan π- ·0=—tan 0, ―0=-cot 0. For, let xop be any plane angle 0, and take Q symmetric with then and the distances of P, Q are equal, their abscissas opposite, and ordinates equal, i.e., ... y' r' = y: ", -0=sin 0, cosπ—0——cos 0, tan π-0=-tan 0, and so for the other functions. - = Q. E. D. NOTE. If from some point on ox' a perpendicular fall on op, the theorem may be proved directly from the supplementary angle Pox'. FUNCTIONS OF T +0. THEOR. 7. If be any plane angle, then: sin 0, cos + 0 = cos 0, tan π + 0 =+tan 0, csc 0, sec π+0 sec 0, cot + 0 =+ cot 0. For, let xop be any plane angle 0, and take Q symmetric with and the distances of P, Q are equal, their abscissas opposite, and ordinates opposite, i.e.,.r'r, x'=-x, y' =-y, always, i.e., .. y' r'-y: r, x':' = −x:r, y' x'-y-x, sin +0=sin 0, cos π+0=-cos 0, tanπ+0=tan 0, and so for the other functions. Q. E. D. NOTE. Since + is the supplement of 0, the theorem follows directly from ths. 5, 6. FUNCTIONS OF THE COMPLEMENT OF AN ANGLE. THEOR. 8. If be any plane angle, then: sin co-0= cos 0, cos co-0: = csc co-0 = sec 0, sec co-0 sin 0, tan co-0=cot 0, For, let xop be any plane angle ; take o symmetric with P as to the bisector of the first quarter, so that yoQ = − 0 ; then ... and Y' the abscissa of q is equal to the ordinate of P, and the ordinate of Q to the abscissa of P, i.e., •.• r'=r, x'=y, yx, always, NOTE. The word cosine stands for complement-sine, sine of complement; so for cotangent and cosecant. |