TRIGONOMETRY OF RIGHT TRIANGLES

TRIGONOMETRIC RATIOS Consider a right triangle with as one of its acute angles. The trigonometric ratios are defined as follows. hypotenuse opposite adjacent sin =cos =tan = cot =csc = sec = 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 =cos = tan =cot =csc = sec =

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, c a b B CA

EXAMPLE: 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

EXAMPLE: 1.Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the acute angle A: b) a=1, b=, c=2

EXAMPLE: 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.

EXAMPLE: 3.Find the other five functions of the acute angle A, given that: a) tan A =

EXAMPLE: 3.Find the other five functions of the acute angle A, given that: b) sec A =

EXAMPLE: 3.Find the other five functions of the acute angle A, given that: c) sin A =

FUNCTIONS OF COMPLIMENTARY ANGLES c a b B C A sin A =cos A =tan A =cot A =sec A =csc A =cos B =sin B =cot B =tan B =csc B =sec B = 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=90 0 ), then

FUNCTIONS OF COMPLIMENTARY ANGLES sin B = sin = cos Acos B = cos = sin Atan B = tan = cot Acot B = cot = tan Asec B = sec = csc Acsc B = csc = 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.

EXAMPLE: 4.Express each of the following in terms of its cofunction: a) sin b) csc c) tan

EXAMPLE: 5.Determine the value of that will satisfy the ff.: a) csc = sec 7 b) sin =

TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES 45 0, 30 0 AND 60 0 To find the functions of 45 0, construct a diagonal in a square of side 1. By Pythagorean Theorem this diagonal has length of sin 45 0 =cos 45 0 = tan 45 0 = csc 45 0 =sec 45 0 = cot 45 0 =

To find the functions of 30 0 and 60 0, 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 30 0 and By Pythagorean Theorem the length of the altitude is

sin 30 0 = cos 30 0 = csc 30 0 = 2 tan 30 0 = cot 30 0 = sec 30 0 = sin 60 0 = cos 60 0 = tan 60 0 = cot 60 0 =csc 60 0 = sec 60 0 = 2

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: a) 3 tan sin – cos b) 5 cot tan sin 30 0 c) cos – csc – sec 30 0 d) tan cot 30 0 – sin 60 0 e) tan cot – sin

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: a) 3 tan sin – cos

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: b) 5 cot tan sin 30 0

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: c) cos – csc – sec 30 0

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: d) tan cot 30 0 – sin 60 0

EXAMPLE: 6.Without the aid of the calculator, evaluate the following: e) tan cot – sin

Find…. 1.sin 32 o = 2.cos 81 o = 3.tan 18 o = 4.sec 58 o = 5.cot 78 o =

IF sin  = find  IF cos  = find  IF tan  = find  Use Trigonometry To Find Angles

a  ………………….. c  …………………..

Use trigonometric about special right triangles to find the value of x and y.

Find the missing lengths

*The angle between the HORIZONTAL and a line of sight is called an angle of elevation or an angle of depression Trigonometric Word Problems

A 20-foot ladder is leaning against a wall. The base of the ladder is 10 feet from the wall. What angle does the ladder make with the ground 10 ? 20 A = 60 °

How tall is a bridge if a 6-foot tall person standing 100 feet away can see the top of the bridge at an angle of 60 degrees to the horizon? °

A hot air balloon is flying at an altitude of 1500 m. The angle of depression from the balloon to a landmark on the ground is 30º. a) What is the balloon’s horizontal distance to the landmark, to the nearest metre? b) What is the balloon’s direct distance to the landmark, to the nearest metre?

Two buildings are 30 m apart. The angle from the top of the shortest building to the top of the taller building is 30°. The angle from the top of the shorter building to the base of the taller building is 45°. What is the height of the taller building to the nearest metre?