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Acute Angles and Right Triangles

This page is meant to serve as a quick overview of the basics of acute angles and right triangles.

Right Triangle Based Definitions of the Trigonometric Functions

For any acute angle Θ in standard position, the following applies:

sin Θ = y / r = (opposite side) / (hypotenuse) cos Θ = x / r = (adjacent side) / (hypotenuse) tan Θ = y / x = (opposite side) / (adjacent side)
csc Θ = r / y = (hypotenuse) / (opposite side) sec Θ = r / x = (hypotenuse) / (adjacent side) cot Θ = x / y = (adjacent side) / (opposite side)

Cofunction Identities

For any acute angle Θ in standard position, the following applies:

sin Θ = cos(90-Θ) cos Θ = sin(90-Θ) tan Θ = cot(90-Θ)
csc Θ = sec(90-Θ) sec Θ = csc(90-Θ) cot Θ = tan(90-Θ)

Function Valuse of Special Angles
Θ sin Θ cos Θ tan Θ cot Θ sec Θ csc Θ
30° ( π/6 ) 1 / 2 √3 / 2 √3 / 3 √3 2√3 / 3 2
45° ( π/4 ) √2 / 2 √2 / 2 1 1 √2 √2
60° ( π/3 ) √3 / 2 1 / 2 √3 √3 / 3 2 2√3 / 3

Reference Angle Θ' for Θ in (0°,360°)

Reference Angle - A Reference Angle an acute version of any given angle. In the standard position, the reference angle is the smallest angle between the terminal side and the x-axis. The absolute value of the trig functions of angle Θ are the same as the trig values of the reference angle for Θ.

Θ in Quadrant Θ' is
II 180° Θ
III Θ - 180°
IV 360° - Θ

Procedures for Discovering the Trigonometric Values for any Angle

Step 1 Add or subtract 360° as many times as necessary to get the angle between 0° and 360°.
Step 2 Find the reference angle using the chart above.
Step 3 Calculate the trigonometric values for Θ'.
Step 4 Dtermine the proper sign for the value found in step 3 using ASTC (All Sin Tangent Cos).

Solving Applied Trigonometry Problems

Step 1 Sketch a diagram and lable it with all the givens. Provide the quantity to be found with a variable.
Step 2 Write an equation based on the sketch from Step 1 that relates the variable with the given quantities.
Step 3 Solve for the variable. Check the diagram and ensure the answer makes sense.