Trigonometric Functions of Acute Angles Section 6.1 Trigonometric Functions of Acute Angles Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Determine the six trigonometric ratios for a given acute angle of a right triangle. Determine the trigonometric function values of 30º, 45º, and 60º. Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle. Given the function values of an acute angle, find the function values of its complement.

Right Triangles and Acute Angles An acute angle is an angle with measure greater than 0º and less than 90º. Greek letters such as  (alpha),  (beta),  (gamma),  (theta), and  (phi) are often used to denote an angle. We label the sides with respect to angles. The hypotenuse is opposite the right angle. There is the side opposite  and the side adjacent to . Side opposite  Side adjacent to  Hypotenuse 

Trigonometric Ratios The lengths of the sides of a right triangle are used to define the six trigonometric ratios: sine (sin) cosine (cos) tangent (tan) cosecant (csc) secant (sec) cotangent (cot) Side opposite  Side adjacent to  Hypotenuse 

Trigonometric Function Values of an Acute Angle  Let  be an acute angle of a right triangle. Then the six trigonometric functions of  are as follows:

Example In the triangle shown, find the six trigonometric function values of (a)  and (b) .   13 5

Example In the triangle shown, find the six trigonometric function values of (a)  and (b) .   13 5

Reciprocal Functions Note that there is a reciprocal relationship between pairs of the trigonometric functions.

Example Given that find csc , sec , and cot . Solution:

Example If and  is an acute angle, find the other five trigonometric function values of . Solution: Use the definition of the sine function that the ratio and draw a right triangle. Use the Pythagorean equation to find a. 7 a 6 

Example (cont) Use the lengths of the three sides to find the other five ratios.

Function Values of 45º A right triangle with one 45º, must have a second 45º, making it an isosceles triangle, with legs the same length. Consider one with legs of length 1. 45º 1

Function Values of 30º A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it. 30º 60º 2 1

Function Values of 60º A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it. 30º 60º 2 1

Example As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.) Solution: Draw the situation, label the acute angle and length of the adjacent side.

Example (cont) The balloon is approximately 0.7 mi, or 3696 ft, high.

Function Values of Any Acute Angle Angles are measured either in degrees, minutes, and seconds: 1º = 60´, 1´ = 60´´; referred to as the DºM´S´´ form or are measured in decimal degree form, expressing the fraction parts of degrees in decimal form

Examples Find the trigonometric function value, rounded to four decimal places, of each of the following: Solution: Check that the calculator is in degree mode.

Example A window-washing crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position? Solution: Draw the situation, label the hypotenuse and length of the side adjacent to .

Example (cont) Use a calculator to find the acute angle whose cosine is 0.26: Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground.

Cofunction Identities Two angles are complementary whenever the sum of their measures is 90º. Here are some relationships.  90º – 

Example Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan 18º ≈ 0.3249, find the six trigonometric function values of 72º. Solution: