SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length of the hypotenuse: Steps for solving the formula for c: 1. 2. 3. 4. To find the length of a leg: Steps for solving the formula for a or b: 1. 2. 3. 4. Shortcuts: To find the length of the hypotenuse: To find the length of a leg:
Examples: Find the length of the missing side of each triangle. Write answer as exact and rounded to the nearest hundredth. d) e) f) g) h) i) j)
Distance Formula: Example: Find the distance between ( 1,5) and ( ) 7, 1. 1. Plot the two points on a graph and 2. Draw a right triangle with connect them with a segment. your segment as the hypotenuse. 3. Figure out the lengths of the legs. 4. Plug into the Pythagorean Theorem. -or- Use the distance formula (but be careful with your negatives!) ( ) ( ) 2 2 d = x x + y y 2 1 2 1
Examples: Find the distance between each set of points. d) ( 5, 6 ) and ( 1, 2) e) ( 4, 7 ) and ( 9, 3) f) ( 2,3 ) and ( 5, 7)
SM 2 Date: Section: Objective: Review: How do you find a missing side length in a right triangle? Use the to find the length of the hypotenuse for each right triangle. Express your answers in simplest radical form. There are 2 common types of right triangles with special patterns that can be found using the Pythagorean Theorem. 45-45 -90 Right Triangles: Legs are Hypotenuse= Examples: Find the value of each variable.
d) e) f) g) h) 30-60 -90 Right Triangles: Hypotenuse= Long Leg= Examples: Find the value of each variable.
d) e) f) g) h) i) j) k) l)
SM 2 Date: Section: Objective: Trigonometry: Trigonometric Ratio: 3 main or most common trigonometric ratios: 1) sinθ = 2) cosθ = 3) tanθ = Adjacent means next to θ (the angle you are focusing on) Opposite means across from θ (the angle you are focusing on) Hypotenuse is the side across from the right angle A common way to remember this is: Steps to find the trigonometric ratios 1) 2) 3) 4)
Examples: Find the exact values of sin θ, cos θ, and tan θ. No matter how big the triangle is, the values of the trigonometric functions for a certain size angle will remain x y z the same. For example, in the diagram below, tan 27 = = =. The value of the tangent is the same in all a b c three triangles even though they are different sizes. The same is true for the sine and cosine. Examples: Use a calculator to approximate each value to four decimal places. Make sure your calculator is in degree mode. a) sin 27 b) cos 27 c) tan 27 Steps to find the missing side of a right triangle if you know an angle and one other side: 1) 2) 3) 4)
Examples: Write an equation involving sine, cosine, or tangent that can be used to find the missing length. Then solve the equation. Round your answers to the nearest tenth. d) e) f) g) h) i) j) k) l)
Examples: Draw and label a triangle to illustrate the situation. Find the length of the missing side and give the values of the requested functions. a) Given 7 sinθ =, find tanθ and cosθ. 25 b) Given 15 tanθ =, find sinθ and cosθ. 8
SM 2 Date: Section: Objective: Inverse functions: When do you use inverse functions? sin ( ANGLE ) cos( ANGLE ) tan ( ANGLE) = RATIO OF SIDES 1 sin ( RATIO OF SIDES) = ANGLE = RATIO OF SIDES 1 cos ( RATIO OF SIDES) = ANGLE = RATIO OF SIDES 1 tan ( RATIO OF SIDES) = ANGLE Examples: Find the measure of the indicated angle to the nearest tenth of a degree.
Solving a Triangle: Figuring out the lengths of all three sides and the measures of all three angles of a triangle. DRAW AND LABEL A TRIANGLE (Label the pieces you know with numbers and the pieces you don t know with vairables)! Decide which of these choices your triangle is like. Then follow the instructions. Show work! If you know the measure of one angle and the length of one side, o To find the measure of the other angle: o To find the lengths of the other sides: If you know the lengths of two of the sides, o To find the length of the other side: o To find the measures of the angles: Examples: Solve ABC. Round answers to the nearest tenth. Show all your work. a) b)
c) d) e) m A = 72, c = 10 f) m B = 20, a = 15 g) b = 4, c = 9 h) a = 7, b = 5
Examples: 1) Draw and label a triangle to illustrate the situation. 2) Find the length of the missing side. 3) Give the values of the requested functions. 4) Give the measure of the angle to the nearest tenth of a degree. a) Given 4 sinθ =, find tanθ and cosθ. 5 b) Given 12 tanθ =, find sinθ and cosθ. 5