Topic 1 Pythagorean Theorem and SOH CAH TOA Unit 3 Topic 1

Before we start… Trigonometry means triangle measurement. Trigonometry is used to measure distances that are difficult to measure directly. ▫ the distance across a canyon, the height of a mountain, how to land an airplane, or the length of a tunnel Trigonometry is used to calculate unknown measures when you know: ▫ the length of two sides ▫ the length of one side and the measure of one angle

Before we start… The triangle to the right is called. Triangles have 3 angles and 3 sides. ▫ We label the vertices with CAPITAL LETTERS. ▫ We label the sides with lower-case letters. ▫ Side c is the LONGEST side, across from the right angle (90˚) and is called the hypotenuse. ▫ Sides a and b are the legs of the right triangle and are called t he adjacent and opposite sides. The sum of the angles in any triangle is 180 . A B C θ b (opposite) a (adjacent) c (hypotenuse)

Before we start… Label: a) the sides, using b) the adjacent side, the opposite side, lower-case letters. and the hypotenuse, given reference  B. A B C θ E G F

Before we start… Label: a) the sides, using b) the adjacent side, the opposite side, lower-case letters. and the hypotenuse, given reference  B. A B C θ E G F e g f The lower-case letters are written across from their upper case counterparts. opposite hypotenuse adjacent The hypotenuse is across from the right angle, the opposite side is across from the reference angle, and the opposite side is the one left over.

Information When two side lengths of a right triangle are known, we can use the Pythagorean Theorem,, to solve for the length of the missing side.

Substitute known values into the formula, keeping in mind that the c value is reserved for the hypotenuse. Example 1 Determining the length using Pythagorean Theorem Determine the missing lengths for the following triangles. Round to the nearest tenth. (a-d). a) Try this on your own first!!!! A B C 5 cm 8 cm Solution: Take the square root of both sides.

The c value is reserved for the hypotenuse. Example 1 Determining the length using Pythagorean Theorem Determine the missing lengths for the following triangles. Round to the nearest tenth. (a-d). b) Try this on your own first!!!! Solution: A B C 12 m 22.3 m

Substitute known values into the formula, keeping in mind that the hypotenuse (17) must be put into the formula as c. Example 1 Determining the length using Pythagorean Theorem Determine the missing lengths for the following triangles. Round to the nearest tenth. (a-d). c) Try this on your own first!!!! Solution: Take the square root of both sides. A C B 17 8 Subtract 64 from both sides.

The hypotenuse, 32, must be put into the formula as c. Example 1 Determining the length using Pythagorean Theorem Determine the missing lengths for the following triangles. Round to the nearest tenth. (a-d). a) Try this on your own first!!!! Solution: A B C

Example 2 Determine the angle Determine the missing angles in the following triangles. a) b) c) Try this on your own first!!!! 45  x 62  x 80  25  30  x

Example 2 Determine the angle Remember: All the angles in a triangle must add up to 180 ⁰. When we are given 2 of the 3 angles, we can subtract them both from 180 ⁰ to see what we will have left for the third. a) b) c) Try this on your own first!!!! 45  x 62  x 80  25  30  x

Information Sometimes we are only given one angle and one side in a right triangle and asked to find a missing angle or side. How do we do this? Can we use Pythagorean Theorem? Do you remember SOH CAH TOA?

Information SOH CAH TOA is the word that we use to remember what to do with right triangles to solve for either a missing angle (when given two sides), or a missing side (when given one angle and one side). A B C θ

Information SOH CAH TOA is the word that we use to remember these ratios.

Information To find a missing side using the trig ratios: Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle. Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side. Substitute all known values into the equation, including the variable for the unknown side length. Cross multiply and divide to determine the unknown side length.

Example 3 Determine the length of the missing side. Round all answers to the nearest tenth. a)b)c) Try this on your own first!!!! 20  52 cm x 47  19 m x 12 m 53  x

 Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.  Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side.  Substitute all known values into the equation, including the variable for the unknown side length.  Cross multiply and divide to determine the unknown side length. Example 3: Solution 20  52 cm x a) opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the missing side is the adjacent side, and a given side is the hypotenuse, we select the cosine ratio.

47  19 m x  Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.  Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side.  Substitute all known values into the equation, including the variable for the unknown side length.  Cross multiply and divide to determine the unknown side length. Example 3: Solution b) opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the missing side is the opposite side, and a given side is the adjacent, we select the tangent ratio.

12 m 53  x  Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.  Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side.  Substitute all known values into the equation, including the variable for the unknown side length.  Cross multiply and divide to determine the unknown side length. Example 3: Solution c) opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the missing side is the hypotenuse, and a given side is the opposite, we select the sine ratio.

Example 4 Determining the length A ladder resting on the side of a building forms an angle of 44  with the ground. If the ladder is touching the building at a height of 8 m from the ground, how long is the ladder, to the nearest tenth? Try this on your own first!!!!

Example 4: Solution 44 ⁰ Draw a triangle and label what you know and what you are going to solve for. x 8m hypotenuse opposite Based on the reference angle, we have the opposite side and the hypotenuse. This means that we can use the sine ratio.

Information To find a missing angle using the trig ratios: Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle. Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides. Substitute all known values into the equation, including the variable for the unknown angle measure. Use the inverse function (sin -1, cos -1, or tan -1 ) to find the angle.

Example 5 Determining the angle Determine the measure of the missing angle. Round all answers to the nearest degree (whole number). a)b)c) Try this on your own first!!!! x 10 m 8 m 20 cm 30 cm x x 2 m 5 m

Example 5: Solution a) x 10 m 8 m  Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle.  Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides.  Substitute all known values into the equation, including the variable for the unknown angle measure.  Use the inverse function (sin -1, cos -1, or tan -1 ) to find the angle. opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the two sides we are given are opposite and adjacent, we can use the tangent ratio.

Example 5: Solution b)  Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle.  Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides.  Substitute all known values into the equation, including the variable for the unknown angle measure.  Use the inverse function (sin -1, cos -1, or tan -1 ) to find the angle. opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the two sides we are given are adjacent and hypotenuse, we can use the cosine ratio. 20 cm 30 cm x

Example 5: Solution c)  Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle.  Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides.  Substitute all known values into the equation, including the variable for the unknown angle measure.  Use the inverse function (sin -1, cos -1, or tan -1 ) to find the angle. opposite hypotenuse adjacent Remember: the opposite side is across from the angle that we are dealing with, and the hypotenuse is across from the right angle. Remember: Since the two sides we are given are opposite and hypotenuse, we can use the sine ratio. x 2 m 5 m

Example 6 Determining the Angle A 56 m wire, supporting a TV tower 45 m tall, joins the top of the tower to an anchor point on the ground. What is the angle, , to the nearest degree, that the wire makes with the ground? Try this on your own first!!!!

Example 6: Solution opposite hypotenuse Based on the reference angle, we have the opposite side and the hypotenuse. This means that we can use the sine ratio.

Need to Know: Solving for a missing side in a right triangle: When two sides of a right triangle are known, we can use the Pythagorean Theorem, to determine the third side. If an angle and only one side are known: ▫ Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle. ▫ Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side. ▫ Substitute all known values into the equation, including the variable for the unknown side length. ▫ Cross multiply and divide to determine the unknown side length.

Need to Know: Solving for a missing angle in a right triangle: When two angles of a right triangle are known, we can use the triangle sum theorem to find the third angle. If only sides are given: ▫ Label the sides as opposite, adjacent, and hypotenuse, with respect to the unknown angle. ▫ Determine which trig ratio (sin, cos, or tan) would include the unknown angle and the two given sides. ▫ Substitute all known values into the equation, including the variable for the unknown angle measure. ▫ Use the inverse function (sin -1, cos -1, or tan -1 ) to find the angle. You’re ready! Try the homework from this section.