Solving Equations with Trig Functions

Labeling a right triangle A

Using Pythagorean Theorem If only 2 sides of the triangle are given, use the Pythagorean Theorem to solve for the missing side a 2 + b 2 = c 2

Practice Name the side (Opposite or Adjacent) to the given angle. A B C Hypotenuse: Opposite of <C: Adjacent to <C: Opposite of <A: Adjacent to <A:

Practice Name the side (Opposite or Adjacent) to the given angle. L S U Hypotenuse: Opposite of <S: Adjacent to <S: Opposite of <U: Adjacent to <U:

Practice Name the side (Opposite or Adjacent) to the given angle. A B T Hypotenuse: Opposite of <A: Adjacent to <A: Opposite of <B: Adjacent to <B:

Practice Name the side (Opposite or Adjacent) to the given angle. X A E Hypotenuse: Opposite of <X: Adjacent to <X: Opposite of <E: Adjacent to <E:

Trig Functions Trig Ratios of angles of a right triangle relates the sides of the right triangle Sine: sin A = Opposite/Hypotenuse Cosine: cos A =Adjacent/Hypotenuse Tangent: tan A = Opposite/Adjacent

Example (Soh-Cah-Toa) Find all three trig ratios of the following triangle: sin A = cos A = tan A =

Example A B C sin A = cos A = tan A =

Example A = 12 cm, C = 15 cm sin A = cos A = tan A = Find the Hypotenuse

Simplifying Radicals

Perfect Squares

= 2 = = 5 = = This is a piece of cake! Simplify

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

= = = = = Simplify = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

Combining Radicals + To combine radicals: combine the coefficients of like radicals

Simplify each expression

Simplify each expression: Simplify each radical first and then combine.

= = = = = Simplify = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

Warm-Up

Simplify each expression

Multiplying Radicals * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

Multiply and then simplify

Dividing Radicals To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator

That was easy!

This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to RATIONALIZE by multiplying the fraction by something so we can eliminate the radical in the denominator. 42 cannot be simplified, so we are finished.

This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.

This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. Reduce the fraction.

= X = = = = Simplify

= = = =

= = = =

The Unit Circle and Trig Functions

Trig Functions Reminder Trig Ratios of angles of a right triangle relates the sides of the right triangle Sine: sin A = Opposite/Hypotenuse Cosine: cos A =Adjacent/Hypotenuse Tangent: tan A = Opposite/Adjacent

Trigonometric Functions in the Calculator Evaluate each of the following using your calculator (round to the nearest thousandth) sin (62 °)sin(-Ɵ) = - sin(Ɵ) 2. cos (132 °)tan (-Ɵ) = - tan (Ɵ) 3. tan (-87 °)cos (-Ɵ) = cos (Ɵ) 4. cos (178 °) 5. sin(-60 °) 6. tan(95 °)

Inverse Trigonometric Functions in the Calculator Solve the following equations and express your answer in degrees 1. sin (x) = cos (x) = tan (x) = cos (x) = sin (x) = tan (x) = -12.8

Using Trig to Find Angles and Sides Sometimes you will be given a triangle and you’ll need to use trigonometry to solve for an angle in the triangle. When this happens, follow these steps: Label the angle, and figure out which sides are given: Opposite, Adjacent, or Hypotenuse Use the information found in #1 to determine which trig function to use Set up the trig function using a variable in place of the angle you don’t know. Take the Inverse Trig Function of each side of the equation Type the Inverse Trig expression into your calculator to solve. Make sure you’re in the right mode

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. a = 36; b = 40

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. c = 31; b = 28

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. a = 30; c = 37

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. a = 14; b = 22

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. b = 18; c = 36

Examples: Set up a trig ratio with the given sides and the angle. Then, solve for the angle. c = 28; a = 24

To find the missing side given an angle and side: Set up the appropriate trig ratio Make sure your calculator is in degree mode Solve for the missing side

Example: =30, a = 10, find b

Example: =42, a = 6, find c

Example: =71, c = 10, find a

Example: =53, b = 17, find c

Example: =26, a = 5, find b

Example: =46, b = 9, find a

Homework At a point 20 meters from a flagpole, the angle of elevation of the top of the flagpole is 48◦. How tall is the flagpole?

If a rocket flies 2◦ off course for 1000 miles, how far from the correct path will the rocket be?

Warm-Up Problem As it leans against a building, a 9-meter ladder makes an angle of 55◦ with the ground. How far is the bottom of the ladder from the base of the building?

QUIZ TOMORROW

Review Problems Simplify each expression:

Review Problems Simplify this expression √ 12c 4 b 3 d 5 √ 2c 3 b 3 d 2

Angles of Elevation and Depression

Angle of Elevation/ Depression

Example The sun casts a shadow 172 ft. long off a building that is 125 ft. tall. What is the angle of depression of the sun?

Example Mike is taking a picture of a building. The angle of elevation from his camera to the top of the building is 22 . If his camera is 5ft off the ground and he is standing 300 ft from the building, how tall is the building?

Example The angle of elevation to the top of the Empire State Building in New York is found to be 11  from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Example From the top of a 200-ft lighthouse, the angle of depression to a ship in the ocean is 23 . How far is the ship from the base of the lighthouse?

Example A water tower is located 325 feet from a building. From a window in the building it is observed that the angle of elevation to the top of the tower is 39  and the angle of depression to the bottom of the tower is 25 . (see the figure below) How tall is the tower? How high is the window?

Homework YOU HAVE A HOMEWORK ON ALEKS HOLDING OVER FROM LAST WEEK

12. Consider this right triangle along with the given information and answer the questions below: a. What is the measure of the Hypotenuse…..Do NOT ROUND. b.What is the measure of Angle A? c.What is the Sine of Angle B?….. DO NOT ROUND d.What is the Tangent of Angle C?....Do Not Round

13. A six-meter-long ladder leans against a building. If the ladder makes an angle of 60° with the ground, how far up the wall does the ladder reach? How far from the wall is the base of the ladder? Round your answers to two decimal places, as needed.

14. A ladder leans against a building. The foot of the ladder is 6 feet from the building. The ladder reaches a height of 14 feet on the building. Find the angle the ladder makes with the ground.

15. From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º. Find to the nearest foot, the height of the tree.

16. Angle Q is 90◦and Angle R is 30◦ The hypotenuse measures 20 meters. Find the lengths of the legs.

Warm-Up Reminders

17. From the top of a barn 25 feet tall, you see a cat on the ground. The angle of depression of the cat is 40º. How many feet, to the nearest foot, must the cat walk to reach the barn?