Basic Trigonometry

Trigonometry The word Trigonometry is derived from the ancient greek language and means measurement of triangles. “tri” meaning three. “gon” meaning sides. “metron” meaning measure. Trigonometry is the study of triangles. Using trig, you can solve many problems precisely where you may have only been able to approximate before. Trig is used in everything from architecture to engineering.

Timeline… The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early astronomers used Trig to find out the distance of the stars and planets from the Earth.

Parts of a Right Triangle The hypotenuse will always be the longest side, and opposite from the right angle. Imagine that you are at Angle A looking into the triangle. The opposite side is the side that is on the opposite side of the triangle from Angle A. The adjacent side is the side next to Angle A.

Review B A B A For Angle A Hypotenuse This is the Opposite Side This is the Adjacent Side A Adjacent Side Adjacent Side B For Angle B Hypotenuse This is the Opposite Side This is the Adjacent Side A

Trig Functions Sine = sin Cosine = cos Tangent = tan Cosecant = csc Secant = sec Cotangent = cot

Trig Ratios Each of the 3 ratios has a name Hypotenuse Each of the 3 ratios has a name The names also refer to an angle Opposite A Adjacent

SOHCAHTOA B Here is a way to remember how to make the 3 basic Trig Ratios if we are talking about angle A Hypotenuse Opposite A Adjacent 1) Identify the Opposite and Adjacent sides for the appropriate angle SOHCAHTOA means Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent Put the underlined letters to make SOH-CAH-TOA

Examples of Trig Ratios First we will find the Sine, Cosine and Tangent ratios for Angle P. 20 12 Next we will find the Sine, Cosine, and Tangent ratios for Angle Q Q 16 Remember SohCahToa

Activity: Draw a right triangle on your paper. Use your angle ruler to ensure you get as close to exactly a 90 degree angle as you can. Pick one of the other angles and label it Angle A. Determine which side of your triangle is the hypotenuse, opposite and adjacent. (Label these) Measure (cm) the length of the hypotenuse, opposite side and the adjacent side in cm. Jot these measurements down. Find your three trig ratios. Sin A = ? Cos A = ? Tan A = ? Once you figure these out, write them down on your paper. Also write these ratios as a decimal (round to two decimals) Next, actually measure angle A with your angle ruler and label it. Make sure your calculator is in degree mode if you are using degrees and radian mode if you are using radians, take the sin of whatever angle you measured, does this match your ratio? Do this for Cos and Tan. Jot this down.

Triangle Definitions of other Trig Functions      

SOHCAHTOA β    

WE CAN USE OUR TRIGONOMETRY KNOWLEDGE TO SOLVE FOR A MISSING SIDE LENGTH IF WE KNOW THE ANGLE!!! 1. 2. c a    

SOHCAHTOA Practice

Engineering Connection Sometimes engineers cannot directly measure an object's size or distance because it would take too much time, or it is physically impossible (a tape measure to find the distance from the Earth to Pluto?). Instead of actually measuring a size or distance, engineers use trigonometry and other mathematical relationships to estimate it very accurately.

Activity You need to find the exact distance across the river from your tree (green tape) to the tree directly across the river (partners green tape). HOWEVER… there are rules. Only four of them… 1. You may not measure across the river. (This includes measuring string draped across the river.) 2. You must use trig ratios of some sort to figure out this distance. 3. You may only use the tools you have been given… 4. You may not walk in the river. That is all… may the odds be ever in your favor.