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- Introduction to right triangle trigonometry
- Define three basic trigonometric functions sine,

cosine, and tangent - Using right triangle trigonometry to solve

problems

Trigonometry

- Trigonometry is the study of the relationships

between the sides and the angles of triangles - In any right triangle with the same angle

measure, the ratio of the lengths of the legs is

the same - This follows from AA similarity,
- since all such triangles are similar
- The legs are identified based on where
- they are in relation to the angle
- One leg is the side opposite the angle
- The other leg is the side adjacent to the angle

Trigonometry

- The tangent of an angle in a right triangle is

the ratio of the length of the opposite side to

the length of the adjacent side - If the length of one leg and the measure of one

angle of a right triangle is known, the tangent

provides a method of finding the length of the

other leg - Given the triangle at right, the tangent of 31

is 3/5 - Use the tangent to find the height of the tree in

this picture

Trigonometry

- In addition to the tangent, there are two other

trigonometric functions defined in this chapter - The sine of an angle in a right triangle is the

ratio of the length of the opposite side to the

length of the hypotenuse - The cosine of an angle in a right triangle is the

ratio of the length of the adjacent side to the

length of the hypotenuse - Trig functions are usually abbreviated tan, sin,

and cos - Scientific calculators usually have buttons for

these three trigonometric functions - Before calculators, the values were calculated by

hand, and books were published containing tables

giving the values of trig functions for different

angles

Trigonometry

- The mnemonic SOH-CAH-TOA is sometimes used to

remember the relationships between the sides

of a triangle and trig functions - Consider angle A in the triangle at right
- This is a function, not multiplication!
- This is read as sine of A, cosine of A,

tangent of A - Note that the leg opposite one angle is the leg

adjacent to the other angle

Trigonometry

- Trigonometry provides another way to use indirect

measurement with right triangles - Given a distance and an angle in a right

triangle, the length of any other side can be

found using sine, cosine, or tangent functions - Surveyors use trigonometry to calculate distances

in construction - Example
- Find the length of the hypotenuse of a right

triangle if an acute angle measures 20 and the

side opposite the angle measures 410 feet - First, make a sketch of the problem
- Determine which trig function to use
- SOH-CAH-TOA Sine Opposite/Hypotenuse
- Set up an equation and solve for x

Trigonometry

- If two sides of a right triangle are known,

inverse trig functions can be used to find the

angle measures - Scientific calculators can also find inverse trig

functions - The inverse functions are written as sin-1x,

cos-1x, and tan-1x - If sinA x, then A sin-1x
- For example, to find the angle with a sine of ½
- sinA ½ A sin-1(½) A 30
- Example
- Use an inverse trig function to find the angle

opposite the shorter leg of a right triangle with

legs of 8 and 15 inches - First, make a sketch of the problem
- Use inverse tangent to find mÐA
- tanA 8/15 A tan-1(8/15) A 28

Problem Solving with Right Triangles

- Two terms used when making observations using

angles - Angle of elevation is the angle looking up from

horizontal - Angle of depression is the angle looking down

below horizontal

- The angle of elevation for the person at left is

equal to the angle of depression for the person

at right - Example
- If the angle of elevation from the sailboat to

the top of the lighthouse is 16, and the

lighthouse is 121 feet tall, how far is the boat

from shore?

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