SOH CAH TOA

What is Trigonometry?

13.1 USE TRIGONOMETRY WITH RIGHT TRIANGLES

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles. Those relationships between the sides and angles of a triangle can be described by the term, trigonometric functions.

As if we were standing at angle A and looking out: Hypotenuse = ______________________________ Adjacent side = _____________________________ Opposite side =_____________________________

RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS

Let be an acute angle of a right triangle. The six trigonometric functions of are defined as follows:

Sine: sin = opposite hypotenuse

Cosine: cos = adjacent hypotenuse

Tangent: tan = opposite adjacent

Cosecant: csc = hypotenuse opposite

Secant: sec = hypotenuse adjacent

Cotangent: cot = adjacent opposite

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Example: Evaluate the six trigonometric functions of the angle theta. #7-8 on HW sin =

csc =

cos =

sec =

tan =

cot =

Example: Let be an acute angle of a right triangle. Find the values of the other five trig. functions of . #9-15 on HW

csc 10 7

sin =

csc 10 7

cos =

sec =

tan =

SPECIAL TRIANGLES

cot =

Example: Use the special angles to find the exact values of x and y. (no decimals) Steps:

1. Determine the type of special angle triangle (45-45-90 or 30-60-90) and insert the known relationships. 2. Identify the given side and set up an equation to solve for x. 3. Solve that equation and apply your answer to find the other sides.

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Calculator MODE: As we begin to work with degrees, please make sure your calculator is in degrees!!!

What does it mean to "solve a right triangle"?

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What can we use to help find the missing sides and angles?

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Example 4: Assuming that the following are right triangles, find the missing side. To do so:

1. Look at the given information and the unknown. Starting with the angle, label the sides as the hyp, opp, or adj.

2. Determine which function (sin, cos, or tan) relates the two sides and the angle. 3. Write an equation modeling this relationship. 4. Use a proportion to solve for the missing side. Round each answer to the nearest hundredth. A = 23?, b = 32 Find c.

Example 5: Assuming the following are right triangles, solve ABC by using the given measurements. Round each answer to the nearest hundredth.

A = 75?, b = 6

Angle of Elevation The angle created when an observer standing on the ground would need to raise his or her line of sight above a horizontal line (ground) in order to see an object. In other words, it's the angle created by following the path of one's sight from the ground to an object.

Angle of Depression

If an observer were UP ABOVE and needed to look down, the angle of depression would be the angle that the person would need to lower his or her line of sight.

The angle of elevation and depression are always congruent because they are alternate interior angles created by two parallel lines, the ground and the horizontal in the sky.

Example 6: Building Height You are measuring the height of your school building. You stand 25 feet from the base of the school. The angle from a point on the ground to the top of the school is 62?. Estimate the height of the school to the nearest foot.

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