TRIGONOMETRY is a branch of Geometry that deals with TRIANGLES Trigonometry can be used to figure out unknown measurements of parts of triangles Why should we learn about Trigonometry?

Figuring out the unknown measurements of ALL of the parts of a triangle is called ‘Solving a Triangle’. Usually, you can ‘solve a triangle’ if you know the measures of three parts of the triangle.

We will be learning three applications of trigonometry: The three ‘trig functions’: Sine, Cosine, and Tangent (abbreviated sin, cos, & tan ) The ‘Law of Sines’ The ‘Law of Cosines’

Which of the three applications is to be used depends on what measurements of a triangle are known: ? 25° ? ° 45° 23 16

The three ‘trig functions’: (Sine, Cosine, and Tangent) only apply to RIGHT triangles The ‘Law of Sines’ and the ‘Law of Cosines’ apply to ALL triangles

Which of the three applications is to be used depends on what measurements of a triangle are known: 12” 18” ? Pythagorean Theorem Since you know two sides of a right triangle, and you want to figure out the third side, you use …

Which of the three applications is to be used depends on what measurements of a triangle are known: 25° ? 16” Trig functions Since you know three pieces of information ( one is that it is a right triangle)

Which of the three applications is to be used depends on what measurements of a triangle are known: 14” 11” 55° Law of Sines Since you know one opposite side / angle pair and something else

Which of the three applications is to be used depends on what measurements of a triangle are known: 19” 115° 23” Law of Cosines Since you know a ‘complete corner’ : two sides and the included angle x

Which of the three applications is to be used depends on what measurements of a triangle are known: 19” xºxº 23” Law of Cosines Since you know three sides and you are trying to find an angle. 28”

The three 'trig functions' (sine, cosine, and tangent) are things that are done to ANGLES, not SIDES. The important angle of the problem is called THETA, Θ. THETA Θ reminds me of an eyeball, and that helps me remember that the angle Θ is the angle that the situation is viewed from. The Three Trig functions

Θ hypotenuse Naming the sides of a right triangle 1) Locate the viewpoint, Θ 2) Locate the hypotenuse 3) Locate the adjacent side and the opposite side adjacent opposite

Θ hypotenuse Naming the sides of a right triangle 1) Locate the viewpoint, Θ 2) Locate the hypotenuse 3) Locate the adjacent side and the opposite side adjacent opposite

Θ Hypotenuse,… or opposite ? Naming the sides of a right triangle 1) Locate the viewpoint, Θ 2) Locate the hypotenuse 3) Locate the adjacent side and the opposite side adjacent Opposite,… or another adjacent ? The viewpoint is NEVER at the right angle !!

The SINE (abbreviated sin ) is the ratio of the opposite side divided by the hypotenuse. 12 ” 20 ” 16 ” Θ º The sine of the 36.87º angle is 12 / 20, which = 0.6. Using your calculator, sin 36.87º =

The COSINE (abbreviated cos ) is the ratio of the adjacent side divided by the hypotenuse. 12 ” 20 ” 16 ” Θ º The cosine of the 36.87º angle is 16 / 20, which = 0.8. Using your calculator, cos 36.87º = , =

The TANGENT (abbreviated tan ) is the ratio of the opposite side divided by the adjacent side. 12 ” 20 ” 16 ” Θ º The tangent of the 36.87º angle is 12 / 16, which = Using your calculator, tan 36.87º =

So, based upon the location and viewpoint of the acute angle, called “ Θ ” : Sin Θ = opposite / hypotenuse Cos Θ = adjacent / hypotenuse Tan Θ = opposite / adjacent Here are two mnemonics to help you remember these relationships: SOH CAH TOA (which sounds like a cool Native American-ish word ) Sally Can Tell Oscar Has A Hat On Always

67° x 24” How can you determine the length of the side marked x ? 1)From the viewpoint of the angle Ө, the x side is the opposite side, and the 24” side is the hypotenuse. 2)The trig function that relates the opposite side and the hypotenuse is the sine 3) Sin Ө = opp / hyp 4) Sin 67º = x / 24 5) = x / 24 6) (24) (0.9205) = x 7) (24) (0.9205) = Ө ”

x 24” How can you determine the measure of angle x ? 1)From the viewpoint of the angle x, the 18” side is the adjacent side, and the 24” side is the hypotenuse. 2)The trig function that relates the adjacent side and the hypotenuse is the cosine 3) Cos Ө = adj / hyp 4) Cos x º = 18 / 24 5) Cos x º = ) What angle has a cosine = ? 7) Cos -1 (0.7500) = º Ө 18” 41.4 º

To ‘solve a triangle’ using the three trig functions, 1.Draw and label a picture, including the sides and the angle. Use a variable to represent the unknown amount. 2.Identify which two sides are labeled, using the terms ‘opposite’, ‘adjacent’ and ‘hypotenuse’. Use the viewpoint of the angle, theta. 3.Determine which of the three trig functions deal with that pair of side names. 4.Write the proper equation, beginning with the guide equation. Then, substitute the values you know for the sides and angles. 5.Solve for the variable using algebra.

For example: A man in a boat sees the top of the Barnegat Lighthouse at an angle of 20 degrees above the horizon. He knows that the lighthouse is 165 feet tall. How far away from the shore is the boat?

165' 20° The question: how long is this side ? This problem models a right triangle, and we can solve it using trigonometry ! Answer: = 165' / tan 20° =165' /.364 =453 feet THE BOAT IS ABOUT 453 FEET FROM THE SHORE. tan Ө = opp / adj adj = opp / tan Ө

In professional football, a net is raised in front of the bleachers before a field goal or extra point is attempted. The net is hung 25 yards from the end of the end zone. A groundskeeper on the 30-yard line sights the top of the net at a 42 degree angle with the ground. How high is the top of the net?

65 yards = yards x 42°

In professional football, a net is raised in front of the bleachers before a field goal or extra point is attempted. The net is hung 25 yards from the end of the end zone. A groundskeeper on the 30-yard line sights the top of the net at a 42 degree angle with the ground. How high is the top of the net? 65 yards x 42° Answer :Tan Θ = opp / adjTan 42° = x / = x / 65 x = (.9004) (65) x = 58.5 The net is 58.5 yards, or feet, above the groundskeeper’s eyes. So the net is about 181 feet above the ground.

Summary Identify the viewpoint angle, Θ Use SOH CAH TOA, or some other way to remember the side ratios. Write out the correct trig equation Substitute the values for the variables, and solve for the unknowns using algebra If you have to figure out the measure of the angle, use the inverse trig functions, sin -1, cos -1, tan -1 (on most calculators, use the 2 nd button)