5.2 Right Triangle Trigonometry
Example of Evaluating Trigonometric Functions with a Graphing Calculator Degree or Radians
*Note a calculator only finds approximate values so you still need to use pictures of right angles to find functions of 30°, 45°, and 60° angles.
How do you Find the Function Values for the Special Angle 45° or (π/4)?
1. Construct a right triangle with a 45° angle. *The triangle actually has two 45° angles, thus the triangle is isosceles, meaning it has two sides of the same length. 2. Assume each leg of the triangle has a length equal to 1. Then find the length of the hypotenuse using the Pythagorean Theorem. 3. Now Evaluate the Trigonometric Functions of 45° and rationalize the denominators as needed.
How do you Evaluate Trigonometric Functions?
1. Find any missing values for the lengths of all three sides of the triangle (a,b, and c) using the Pythagorean Theorem a^2 + b^2 = c^2. 2. Apply the definitions of the six trigonometric functions of θ. sin θ= opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent csc θ = hypotenuse/adjacent (reciprocal of sine) sec θ = hypotenuse/adjacent (reciprocal of cosine) cot θ = adjacent/opposite (reciprocal of tangent) For the triangle in the image, the six trigonometric functions are: sin θ = 5/13 cos θ = 12/13 tan θ = 5/12 csc θ = 5/13 sec θ = 13/12 cot θ = 5/12 *Note we had to find the value of c (the hypotenuse) using the pythagorean theorem before we could apply the definitions of the six trigonometric functions to evaluate the trigonometric functions.
How do you Evaluate Trigonometric Functions of 30 degrees (or π/6 radian) and 60° (or π/3 radian) angles?
1. Use a right triangle. *Form it by drawing an equilateral triangle-a triangle with all sides the same length. 2. Assume each side has a length equal to 2. 3. Take half of the equilateral triangle to obtain a right triangle. *The right triangle has a hypotenuse of length of 2 and a leg of length 1. The other leg has length a, which can be found using the Pythagorean Theorem. a^2 + 1^2 = 2^2 a^2 + 1 = 4 a^2 = 3 (subtract one from both sides) a= (square root of 3 under the radical sign) take the square root of both sides using the square root property. 4. Now with the right triangle and values of all sides, we can determine the trigonometric functions for 30 and 60 degrees. *See the next card*
What are Cofunctions of Complements?
Any pair of trigonometric functions f and g for which f(θ) = g(90° - θ) and g(θ) = f(90° - θ) are called cofunctions. Using figure 5.27, we can show that the tangent and cotangent are also cofunctions of each other. So are the secant and cosecant.
How do you Determine the Angle of Elevation or θ?
Begin with the tangent function tan θ = side opposite θ/side adjacent θ and a calculator in degree mode to find θ.
What are Inverse trig functions?
Inverse trig functions do the opposite of the "regular" trig functions. For example: Inverse sine (sin^-1 does the opposite of the sine) Inverse cosine (cos^-1) does the opposite of cosine. Inverse tangent (tan^-1) does the opposite of the tangent. Therefore, if you know the trig ratio but not the angle, you can use the corresponding trig function to find the angle.
Right Triangle Definitions of Trigonometric Functions
Note the ratios in the second column in the box are reciprocals of the corresponding ratios in the first column.
How to Use a Calculator to Evaluate Trigonometric Functions
Remember the Reciprocal Identities. To find the value of 30° , set the calculator to the degree mode and enter SIN 30 ENTER on the graphing calculator. To evaluate the cosecant, secant, and cotangent functions, use the key for the respective reciprocal function, SIN, COS, or TAN, and then use the reciprocal key x^-1. *THIS IS NOT THE SAME AS FINDING THE ACUTE ANGLE OF θ*
What are the Inputs and Outputs for Trigonometric Functions?
The inputs for these six trigonometric functions are measures of acute angles in right triangles. The outputs are the ratios of the lengths of the sides of right triangles.
What are the Cofunction Identities?
The value of a trigonometric function of θ equal to the cofunction of the complement of θ. Cofunctions of complementary angles are equal.
What are Trigonometric Identities?
Trigonometric identities describe the many relationships that exist between the six trigonometric functions, such as Reciprocal, Quotient, and Pythagorean Identities.
How to use a Graphing Calculator to Find an Acute Angle of θ in Degrees.
Use the inverse trigonometric key on a calculator. Suppose that sin θ = 0.866. Find θ in the degree mode by using the secondary inverse sine key, usually labelled SIN^-1. SIN^-1 is not a button you push, it is a secondary function for the button labeled SIN.
What are the 6 Reciprocal Identities?
When finding cot θ using tan θ, use the solution you found before rationalizing the denominator or you'll have to rationalize the denominator again *if rationalizing the denominator was necessary*
Does a Particular Acute Angle always give the same ratio of opposite to adjacent sides?
Yes.
