Solving Right Triangles How do you solve right triangles?

The Trigonometric Functions we will be looking at SINE COSINE TANGENT

The Trigonometric Functions SINE COSINE TANGENT

SINE Pronounced “sign”

Pronounced “tan-gent”

COSINE Pronounced “co-sign”

Represents an unknown angle Greek Letter q Pronounced “theta” Represents an unknown angle

What is Trigonometry? Trigonometry is the study of how the sides and angles of a triangle are related to each other. It's all about triangles! Direct quotes from the Trigonometry tutorial at David@catcode.com.

Right Triangle Hypotenuse Opposite q A Adjacent Here’s a basic right triangle. Remember, the line segment opposite the right angle in a right triangle is called the hypotenuse. Since Trig is all about the relationships of the sides and angles of triangles, let’s “do some Trig” and look at the angle A. The line segment opposite the angle A is called ??? The line segment closest to the angle A is called ??? So, we’ve given this triangle’s sides differing names. What about if we chose another angle? Adjacent

Same Right Triangle – Different Angle B q Hypotenuse Adjacent Here we have the exact same triangle, but this time we’re trying to find out relationships between the sides and the angle B. Once again, the side opposite the angle B is called ??? And, the side closest to angle B is called ??? That tells us that the angle determines which side is the “opposite” side and which side is the “adjacent” side. This information is used in Trig to determine sine, cosine, tangent, cosecant, secant and cotangent. Opposite

Trig Definitions: Sine = opposite/hypotenuse Cosine = adjacent/hypotenuse Tangent = opposite/adjacent Cosecant = hypotenuse/opposite Secant = hypotenuse/adjacent Cotangent = adjacent/opposite These six trigonometric functions are the 6 different ratios that you can set up from a right triangle.  (per PowerPoint presentation by Sally Keely, Trigonometric Functions Defined)

x,y r y O How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem. x r = x + y 2

Definitions of Trig Functions Sin = y / r Cos = x / r Tan = y / x Csc = r / y Sec = r / x Cot = x / y O O O O O Here is a different way of looking at the six trigonometric functions. It’s the same as SOH/CAH/TOA, but applied to the x-y axis. These definitions need to memorized, but do you see some similarities? Sin and cos both have r as their denominator. Sin shows the relationship of y to r and cos shows the relationship of x to r. Tan shows the relationship of y to x. What about Csc, Sec and Cot? Do you see that Csc is, again, the reciprocal of sin? Sec is, again, the reciprocal of cos? And, Cot is, again, the reciprocal of tangent? O

The Unit Circle Radius = 1 Now, what if we applied the definitions of the trig functions to this most basic circle.

Graphic from http://home.alltel.net/okrebs/page72.html Can you see that if you placed all triangles with the same r value, or radius, and differing angle values from 1 degree to 360 degrees you would get a circle if you connected all the “x,y” points? “We could generalize this to say that the circle of radius r describes the collection of all triangles with hypotenuse r . . . Any right triangle is similar to some right triangle with hypotenuse length 1.” (per http://mathforum.org/library/drmath/view/53944.html) In making the radius equal to 1, you’ve created what’s known as the unit circle.

30 30, 60, 90 Is a special kind of triangle. y x,y 1 1/2 x √3/2 2 How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem. √3/2 r = x + y 2

45 45, 45, 90 Is a special kind of triangle. y x,y 1 √/2/2 x √2/2 2 How can we apply our knowledge of right triangles to the Cartesian coordinate system? In Trig, we use Greek letters as general terms to stand for the measures of angles. I used the Greek letter, theta, in this diagram. Can you see that the angle, theta, is drawn in standard position on the x-axis and terminates at the point (x,y)? Since Trig is the study of triangles, how could we make this angle into a right triangle? A line segment perpendicular to the x-axis drawn from point (x,y ) ending at the x-axis could serve as the second side of the triangle. Let’s name this side y. A line segment on the x-axis from the point of origination to the line segment y could be the third side of the triangle. We’ll call this side x; thereby, creating a right triangle! Let’s call the hypotenuse of the triangle r. If we knew the length of line segment x and the length of line segment y, how could we compute the length of line segment r in our right triangle? Who remembers the Pythagorean Theorem? X squared plus y squared equals r squared. If we knew the length of line segment x and line segment y, we could compute r by utilizing the Pythagorean theorem. √2/2 r = x + y 2

Finding sin, cos, and tan. Just writing a ratio.

37 35 12 Find the sine, the cosine, and the tangent of theta. Give a fraction. 37 35 12 Shrink yourself down and stand where the angle is. Now, figure out your ratios.

24.5 8.2 23.1 Find the sine, the cosine, and the tangent of theta Shrink yourself down and stand where the angle is. Now, figure out your ratios.

Solving Right Triangles For Angles (“Theta”)

If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle.

Calculating Angle Measures from Trigonometric Ratios Example 4 Use your calculator to find each angle measure to the nearest tenth of a degree. A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71) cos-1(0.87)  29.5° sin-1(0.85)  58.2° tan-1(0.71)  35.4°

Inverse trig functions: Ex: Use a calculator to approximate the measure of the acute angle. Round to the nearest tenth. 1. tan A = 0.5 2. sin A = 0.35 3. cos A = 0.64 26.6° 20.5° 50.2°

Using Trig Ratios to Find a Missing SIDE

To find a missing SIDE Draw stick-man at the given angle. Identify the GIVEN sides (Opposite, Adjacent, or Hypotenuse). Figure out which trig ratio to use. Set up the EQUATION. Solve for the variable. To find a missing SIDE

If you see it up high then we MULTIP LY! 1. H Problems match the WS. A Where does x reside? If you see it up high then we MULTIP LY!

If X is down below, The X and the angle will switch… 2. H O Problems match the WS. Where does x reside? If X is down below, The X and the angle will switch… SLIDE & DIVIDE

3. H Problems match the WS. A

Solving Right Triangles For All Six Parts

To solve a right triangle you need….. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To SOLVE A RIGHT TRIANGLE means to find all 6 parts. To solve a right triangle you need….. 1 side length and 1 acute angle measure -or- 2 side lengths

Given one acute angle and one side: To find the missing acute angle, use the Triangle Sum Theorem. The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees. To find one missing side length, write an equation using a trig function. To find the other side, use another trig function or the Pythagorean Theorem: a2 + b2 = c2. Note: c is the longest side of the triangle a and b are the other two sides

Solve the right triangle. Round decimal answers to the nearest tenth. GUIDED PRACTICE Example 1 A Find m∠ B by using the Triangle Sum Theorem. 42o 180o = 90o + 42o + m∠ B 70 48o = m∠ B 48o B Approximate BC by using a tangent ratio. C Approximate AB by using a cosine ratio. tan 42o = BC 70 cos 42o = 70 AB ANSWER 70 tan 42o = BC AB cos 42o = 70 70 0.9004 BC AB 70 cos 42o = The angle measures are 42o, 48o, and 90o. The side lengths are 70 feet, about 63.0 feet, and about 94.2 feet. 63.0 ≈ BC AB 70 0.7431 AB 94.2 34

Solve a right triangle that has a 40o angle and a 20 inch hypotenuse. GUIDED PRACTICE Example 2 Find m∠ X by using the Triangle Sum Theorem. X 180o = 90o + 40o + m∠ X 50o 50o = m∠ X 20 in Approximate YZ by using a sine ratio. sin 40o = XY 20 20 ● sin 40o = XY 40o Y 20 ● 0.6428 ≈ XY Z 12.9 ≈ XY Approximate YZ by using a cosine ratio. cos 40o = YZ 20 ANSWER 20 ● cos 40o = YZ The angle measures are 40o, 50o, and 90o. The side lengths are 12.9 in., about 15.3 in., and 20 in. 20 ● 0.7660 ≈ YZ 15.3 ≈ YZ 35

Solve the right triangle. Round to the nearest tenth. Example 3 P° + R° + Q° = 180° P° = 180-90-53 P° = 37 37° 24.0 18.1

Solve the right triangle. Round decimals to the nearest tenth. Example 5 Example 4

Solving Right Triangles Example 6 Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: By the Pythagorean Theorem, Method 2: RT2 = RS2 + ST2 (5.7)2 = 52 + ST2 Since the acute angles of a right triangle are complementary, mT  90° – 29°  61°. , so ST = 5.7 sinR. Since the acute angles of a right triangle are complementary, mT  90° – 29°  61°.

Solve the right triangle. Round decimals the nearest tenth. Example 7 Use Pythagorean Theorem to find c… 3.6 Use an inverse trig function to find a missing acute angle… 56.3° Use Triangle Sum Theorem to find the other acute angle… 33.7°

Solve the right triangle. Round decimals to the nearest tenth. Example 8 Pythagorean Theorem A2 + b2 = c2 Where c is the hypotenuse.

Solve the right triangle. Round decimals to the nearest tenth. Example 9

Trig Application Problems MM2G2c: Solve application problems using the trigonometric ratios.

Depression and Elevation horizontal angle of depression line of sight angle of elevation horizontal Discuss alternate interior angles & how top angle makes a 90 degree angle

9. Classify each angle as angle of elevation or angle of depression.

Example 10 Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation to the nearest degree? 5280 feet – 1 mile

Example 11 The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder? Round to the nearest whole number.

Example 12 Find the angle of elevation to the top of a tree for an observer who is 31.4 meters from the tree if the observer’s eye is 1.8 meters above the ground and the tree is 23.2 meters tall. Round to the nearest degree.

Example 13 A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building? Round to the nearest degree.

Example 14 A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck, to the nearest degree?

Example 15 A 5ft tall bird watcher is standing 50 feet from the base of a large tree. The person measures the angle of elevation to a bird on top of the tree as 71.5°. How tall is the tree? Round to the tenth.

Example 16 A block slides down a 45 slope for a total of 2.8 meters. What is the change in the height of the block? Round to the nearest tenth.

Example 17 A projectile has an initial horizontal velocity of 5 meters/second and an initial vertical velocity of 3 meters/second upward. At what angle was the projectile fired, to the nearest degree?

Example 18 A construction worker leans his ladder against a building making a 60o angle with the ground. If his ladder is 20 feet long, how far away is the base of the ladder from the building? Round to the nearest tenth.