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This document defines trigonometric ratios using a right triangle with an acute angle . It defines sine, cosine, tangent, cotangent, secant, and cosecant ratios. It also discusses reciprocal functions and the Pythagorean theorem. Functions of complementary angles are defined, and specific examples are worked out for 45, 30, and 60 angles using triangles and the Pythagorean theorem. The document concludes with examples of calculating trigonometric functions for given sides of a right triangle and for an angle in standard position, as well as evaluating expressions without a calculator.

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TRIGONOMETRIC RATIOS Consider a right triangle with ? as one of its acute angles. The trigonometric ratios are defined as follows . opposite sin ? = hypotenuse ? adjacent opposite adjacent cos ? = hypotenuse hypotenuse sec ? = adjacent opposite tan ? = adjacent hypotenuse hypotenuse csc ? = opposite adjacent cot ? = opposite Note: The symbols we used for these ratios are abbreviations for their full names: sine, cosine, tangent, cosecant, secant and cotangent.
RECIPROCAL FUNCTIONS The following gives the reciprocal relation of the six trigonometric functions. sin ? = 1 csc? cos ? = 1 sec? 1 tan ? = cot ? csc ? = 1 sin? sec ? = 1 cos ? 1 cot ? = tan?
THE PYTHAGOREAN THEOREM The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In symbol, using the ABC as shown, B a C c 2 ? a2 ? b2 c b A
FUNCTIONS OF COMPLIMENTARY ANGLES a a sin A = c cos B = c b cos A = c b sin B = c a tan A = b a cot B = b b cot A = a b tan B = a b sec A = c b csc B = c c csc A = a c sec B = a B a C c b A Comparing these formulas for the acute angles A and B, and making use of the fact that A and B are complementary angles (A+B=900), then
FUNCTIONS OF COMPLIMENTARY ANGLES sin B = sin (900 ? A) = cos cos B = cos (900 ? A) = sin A tan B = tan (900 ? A) = cot A cot B = cot (900 ? A) = tan A sec B = sec (900 ? A) = csc A csc B = csc (900 ? A) = sec A The relations may then be expressed by a single statement: Any function of the complement of an angle is equal to the co-function of the angle.
To find the functions of 450, construct a diagonal in a square of side 1. By Pythagorean Theorem this diagonal has length of 2 . 1 sin = ? 2 0=1 ? cos 45 2 450 1 450 2 1 450 tan 450 = 1 2 2 2 2 csc 450 = 2 sec 450 = 2 cot 450 = 1
To find the functions of 300 and 600, take an equilateral triangle of side 2 and draw the bisector of one of the angles. This bisector divides the equilateral triangle into two congruent right triangles whose angles are 300 and 600. By Pythagorean Theorem the length of the altitude is 3 . 300 2 3 600 1
sin 300 1 = 2 cos 300 3 = 2 tan 300 1 3 = ? 3 3 cot 300 = 3 sec 300 2 2 3 = ? 3 3 csc 300 = 2 cos 600 1 = 2 sin 600 3 = 2 cot 600 1 3 = ? 3 3 tan 600 = 3 csc 600 2 2 3 = ? 3 3 sec 600 = 2
1. Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the acute angle A: a) a=5 , b=12 , c=13 b) a=1 , b= 3 , c=2 2. The point (7, 12) is the endpoint of the terminal side of an angle in standard position. Determine the exact value of the six trigonometric functions of the angle.
3. Without the aid of the calculator, evaluate the following: a) 3 tan2 600 + 2 sin2 300 C cos2 450 b) 5 cot2 450 + 5 tan 450 + sin 300 c) cos2 600 C csc2 300 C sec 300 d) tan 600 + 2 cot 300 C sin 600 e) tan5 450 + cot2 450 C sin4 600
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  • 2. TRIGONOMETRIC RATIOS Consider a right triangle with ? as one of its acute angles. The trigonometric ratios are defined as follows . opposite sin ? = hypotenuse ? adjacent opposite adjacent cos ? = hypotenuse hypotenuse sec ? = adjacent opposite tan ? = adjacent hypotenuse hypotenuse csc ? = opposite adjacent cot ? = opposite Note: The symbols we used for these ratios are abbreviations for their full names: sine, cosine, tangent, cosecant, secant and cotangent.
  • 3. RECIPROCAL FUNCTIONS The following gives the reciprocal relation of the six trigonometric functions. sin ? = 1 csc? cos ? = 1 sec? 1 tan ? = cot ? csc ? = 1 sin? sec ? = 1 cos ? 1 cot ? = tan?
  • 4. THE PYTHAGOREAN THEOREM The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In symbol, using the ABC as shown, B a C c 2 ? a2 ? b2 c b A
  • 5. FUNCTIONS OF COMPLIMENTARY ANGLES a a sin A = c cos B = c b cos A = c b sin B = c a tan A = b a cot B = b b cot A = a b tan B = a b sec A = c b csc B = c c csc A = a c sec B = a B a C c b A Comparing these formulas for the acute angles A and B, and making use of the fact that A and B are complementary angles (A+B=900), then
  • 6. FUNCTIONS OF COMPLIMENTARY ANGLES sin B = sin (900 ? A) = cos cos B = cos (900 ? A) = sin A tan B = tan (900 ? A) = cot A cot B = cot (900 ? A) = tan A sec B = sec (900 ? A) = csc A csc B = csc (900 ? A) = sec A The relations may then be expressed by a single statement: Any function of the complement of an angle is equal to the co-function of the angle.
  • 7. To find the functions of 450, construct a diagonal in a square of side 1. By Pythagorean Theorem this diagonal has length of 2 . 1 sin = ? 2 0=1 ? cos 45 2 450 1 450 2 1 450 tan 450 = 1 2 2 2 2 csc 450 = 2 sec 450 = 2 cot 450 = 1
  • 8. To find the functions of 300 and 600, take an equilateral triangle of side 2 and draw the bisector of one of the angles. This bisector divides the equilateral triangle into two congruent right triangles whose angles are 300 and 600. By Pythagorean Theorem the length of the altitude is 3 . 300 2 3 600 1
  • 9. sin 300 1 = 2 cos 300 3 = 2 tan 300 1 3 = ? 3 3 cot 300 = 3 sec 300 2 2 3 = ? 3 3 csc 300 = 2 cos 600 1 = 2 sin 600 3 = 2 cot 600 1 3 = ? 3 3 tan 600 = 3 csc 600 2 2 3 = ? 3 3 sec 600 = 2
  • 10. 1. Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the acute angle A: a) a=5 , b=12 , c=13 b) a=1 , b= 3 , c=2 2. The point (7, 12) is the endpoint of the terminal side of an angle in standard position. Determine the exact value of the six trigonometric functions of the angle.
  • 11. 3. Without the aid of the calculator, evaluate the following: a) 3 tan2 600 + 2 sin2 300 C cos2 450 b) 5 cot2 450 + 5 tan 450 + sin 300 c) cos2 600 C csc2 300 C sec 300 d) tan 600 + 2 cot 300 C sin 600 e) tan5 450 + cot2 450 C sin4 600