Geometry Day 58

1. Geometry Day 58 Trigonometry

2. Today’s Objective • Right Triangle Trigonometry • Identify and apply the three basic trigonometric ratios • Sine • Cosine • Tangent • Identify the inverse trig functions

3. What is Trigonometry? • Trigonometry is a branch of mathematics that involves finding out information about right triangles. • (Trigonometry can also involve circles, but that is a subject for Pre-Calculus.) • To do trigonometry, you will need a scientific or graphing calculator. • Make sure your calculator is in ‘degree’ mode.

4. Introduction to Trig functions • Your handout contains a table with columns labeled Sine, Cosine, and Tangent. These are often abbreviated as Sin, Cos, and Tan (but they are pronounced the same way). • Locate these buttons on your calculator. • Don’t worry about what these mean right now, but use your calculator to confirm the values in the table. • In other words, check sin 0 and sin 1 (for example) on your calculator and see if it matches what’s on the table. • Now use your calculator to fill in the missing values on the table. • Also, use your calculator to add two rows: 89.5 and 90. • While doing so, see if you can make any observations about the numbers on the table.

5. Trig and Similarity hyp • Consider a right triangle. • Let’s say we know one of theacute angles. • Any right triangle we draw with an angle of x degrees will be similar to this triangle. Why? • Since all triangles with this set of angles are similar, then their sides will always be in the same ratio. • If I’m standing at angle x, then I can label the three sides as follows: • There is the leg across from me, which I’ll call the opposite leg. • The leg next to me will be the adjacent leg. • We’ll call the hypotenuse, the hypotenuse. opp x adj

6. Trig and Similarity hyp • Note that these side labelsare from the perspective of theangle we’re working with. If wewere standing at the other acuteangle, the opposite and adjacent sides would switch. (The hypotenuse will always be the hypotenuse.) • Remember, any right triangle with an angle measuring x degrees will be similar, and therefore will have proportional sides. • So, if I were to take two of these sides and form a ratio, that ratio will be consistent, no matter how big or small the triangle. opp x adj

7. Trig and Similarity hyp • Ancient mathematicians noticedthis, and calculated what the ratiosof the sides are for the differentpossible acute angles (e.g., 10, 20,36, etc.). • Sine, cosine, and tangent refer to the ratios of specific pairs of sides: opp x adj

8. Trig and Similarity hyp • So sin 40 = .6428 means that, whenever a right triangle has a 40 angle, the ratio of the opposite side to the hypotenuse is .6428. • In the past, these values were written down in tables, and if you needed them, you would have to look them up. Today, these tables are programmed into your calculator. opp x adj

9. Sohcahtoa • Many students have a hard time remembering the ratios. The word ‘sohcahtoa’ can help. • Some old hippie caught another hippie trippin’ on acid. • Feel free to create your own pneumonic device.

10. Observations A • Sine and cosine will always be between zero and one. Why? • Tangent starts very small and grows very large. Why? • What is sin A? • What is cos B? • How are A and B related? • Tangent is opp/adj. Another definition is sin/cos. B C

11. Application • How is this useful? • Solve for x and y. • Ask: • From the perspective of this angle,which sides am I working with? • Which trig function relates those sides? • To prevent rounding errors, solve the equation for the variable before you touch your calculator! x 10 y 25

12. Trig and Special Right Triangles • We know the relationships among sides of two right triangles: • Given this knowledge, what are the following values? 1 30 45 2 1 3 2 60 1

13. Reciprocal Trig Functions A • There are three other trig functions,but they are used less. (In fact, itisn’t necessary to use them at all!) • Cosecant (or csc) • Secant (or sec) • Cotangent (or cot) • In other words, these are the multiplicative reciprocals of sin, cos, and tan, respectively. C B

14. Reciprocal Trig Functions A • Let’s say we wanted to solve for x. • Either of these ratios would beappropriate: • Most calculators don’t have the reciprocal functions, so it’s probably best to use cosine. 28 x 18 C B

15. Inverse Trig Functions A • What if we know the sides of a righttriangle, and need to find an angle? • Every mathematical function hasan inverse. The inverse undoeswhat the function did. • The inverses of sin, cos, and tan, respectively, are sin-1, cos-1, and tan-1. • These are sometimes written as arcsin, arccos, and arctan. • Remember, sin takes an angle and gives a ratio. Sin-1 takes a ratio and gives us the angle. 5 3 B C

16. Inverse Trig Functions A • We would set up the equation inthe same way. • What equation would we set upto solve for A? • To solve for the angle, we use the inverse on the ratio. • Solve for B using this technique. 5 3 B C

17. Solving a Right Triangle A • To solve a right triangle is to findall the measurements of its sidesand angles. • You can solve a right triangle if youknow: • Two side lengths, or • One side length and the measurement of one acute angle. • Solve this right triangle. 2 18 B C

18. Solving a Right Triangle A • Solve this right triangle. 19 12 B C

19. Homework 34 • Workbook pp. 103-104