1. 8.3 Trigonometry

2. Similar right triangles have equivalent ratios for their corresponding sides. These equivalent ratios are called Trigonometric Ratios. For example if we have two triangles that have the following angles of 90, 36.86, and and 53.14 degrees. The side lengths compared to one another will always be in the same proportion, that means as long as the two triangles have all 3 angles congruent, their sides will obviously be different but if you take the legs of one triangle and divide them, the quotient you get back will be identical to the legs of the second triangle divided in the same order. 3 4 5 6 8 10

3. These ratios exist for all right triangles that are similar. The ratios have specific names. They are sine, cosine, and tangent. The names of the ratios identify which side lengths you are comparing. There are 3 other ratios and they are referred to as the “reciprocal trig ratios”, their values are found by taking the reciprocals of sine, cosine, and tangent. The reciprocal ratio of sine (sin) is cosecant (csc) The reciprocal ratio of cosine (cos) is secant (sec) The reciprocal ratio of tangent (tan) is cotangent (cot)

4. How to determine a trig ratio. First you have to be provided with an angle. We then label the sides of the triangle in relation to that angle (hypotenuse, opposite, and adjacent) If we are using angle A, then we label BC as the opposite side, and AB as the adjacent side. B C A HYPOTENUSE BC A opposite adjacent opposite adjacent If we are using angle C, then we label AB as the opposite side, and BC as the adjacent side.

5. BC A HYPOTENUSE opposite adjacent So when we talk about the sin(A) we are indicating the ratio of the opposite leg divided by the hypotenuse (when you do the division the value will show as a decimal on your calculator), that ratio will be exactly the same for all right triangles that have an angle congruent to this triangle’s angle A. SOH-CAH-TOA

6. On your own complete the following question.

7. Once we understand how to identify particular ratios we can use the ratios to find a particular distance or a particular angle measurement. We must use our calculators for this, our calculators have stored in them the ratios of the infinitely many similar triangles that could be created. For example, your calculator has been programmed so that it knows the 6 different ratios that can be created from a triangle having angle measurements of 90, 89, and 1 degrees. It then knows the 6 different ratios that can be created from a triangle having 90, 88, and 2 degrees. And 90, 87, and 3 degrees. All the way up to 90, 45, and 45. But it also has the capabilities of knowing the 6 ratios of a triangle of 90, 40.23, and 49.77, and all triangles with real number angle measurements. We will use our calculator to find a missing side length first. When finding a missing distance/length we must know the numerical value of one of the triangle’s acute angles. opposite hypotenuse Now that we know which sides we have in relation to the known angle, we can now use what trig ratio?

8. Now we know what “w” is, we can now solve the triangle (find all 6 parts)

11. These two triangles are similar, based on their angles we have AA. Thus regardless of what we find their side lengths to be, we will always see that the ratios of opposite/hypotenuse, opposite/adjacent, and adjacent/hypotenuse for both triangles will always be equal. You provide me with any 2 values for either a leg or hypotenuse and we will show that the ratios are always the same.

12. Use trig ratios to find missing angle measurements. As long as you know two sides of a right triangle you should always be able to find an angle measurement using one of the 3 trig ratios. On your calculators you will have to use the sin -1 or cos -1 or tan -1. These are referred to as the inverse trig functions. Which we will type in a ratio of the side lengths and the calculator will spit back an angle measurement that always goes with that ratio.

14. Many real life applications use trigonometry. One of the applications we will investigate the most is indirect measurement. Indirect measurement is the process of determining a measurement without physically measuring it.

15. In a right triangle, the legs are 6 and. What is the length of the hypotenuse and what are the measures of the two non-right (acute) angles? Can this be done using different methods?