Pre-Calculus II 4.3 Right Angle Trigonometry y P=(x,y) y P=(x,y) 1 1 y x x x We construct a right triangle by dropping a line segment from point P perpendicular to the x-axis. So now we can view as the measure of an acute angle in the right triangle. 1 P a g e
SOHCAHTOA (pronounced so-cah-tow-ah) Pythagorean Theorem a b c 2 P a g e
Example Find the value of each of the six trigonometric functions of given the following figure. c a=5 b=12 Example Find the value of each of the six trigonometric functions of given the following figure. c=3 a=1 b 3 P a g e
Special Triangle Relationships An equilateral triangle is a triangle with three equal sides. The three angles of an equilateral triangle are also equal. Each angle measures 60. An isosceles triangle is a triangle with exactly two equal sides. The angles opposite these equal sides are also equal. A scalene triangle is a triangle with all three sides unequal. A 45-45-90 triangle is an isosceles right triangle. The two base angles are each 45, and the last angle is 90. The sum of the angles of a triangle is 180. 45 45 90 Properties of a 30-60-90 triangle. A 30-60-90 triangle is an equilateral triangle cut in half. An equilateral triangle has angle measures 60-60-60, therefore when we divide the top angle in half that measure becomes 30, the altitude creates a 90 angle at the bottom. 30 2 90 60 4 P a g e 1
So how to find the values of the trigonometric functions at So how to find the values of the trigonometric functions at So how to find the values of the trigonometric functions at 5 P a g e
Trigonometric Functions and Complements Two positive angles are complements if the sum of their angles is. For example 70 and 20 are complements because 70 +20 =90. The figure to the left shows a right triangle. Because the sum c 90- a of the angles of any triangle is 180, in a right triangle the sum of the acute angles is 90, thus the acute angles are b complements. If one acute angle is the other must be 90 - From above we can conclude that. If two angles are complements then the sine of one equals the cosine of the other. Because of this relationship the sine and cosine functions are called confunctions of each other. (The name cosine is a shortened form of the phrase complement s sine.) 6 P a g e
Any pair of trig. functions f and g for which confunctions. are called Confunction Identities The value of a trigonometric function of is equal to the confunction of the complement of. Confunctions of complementary angles are equal. Example Find a confunction with the same value as the given expression. 1. 2. 7 P a g e
Applications Line of Sight above Observer Angle of elevation Angle of depression Line of Sight below Observer Horizontal The angle of elevation is the angle from the horizontal line to the line of sight above the observer The angle of depression is the angle from the horizontal line to the line of sight below the observer Example A tower that is 125 feet tall casts a shadow 172 feet. Find the angle of elevation. 8 P a g e
Example The irregular shape is a lake. The distance across the lake is unknown. To find the distance a surveyor took the measurement shown. What is the distance across the lake? (=22 ) a 300 yards 9 P a g e