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Chapter 2 Section 2 – 1 Acute Angles and Right Triangles Trigonometric Functions of Acute Angles Right Triangle – Based Definitions of Trigonometric Functions For any acute angle A in standard position, sin A y Side Opposite r Hypotenuse csc A r Hypotenuse y Side Opposite cos A x Side Adjacent r Hypotenuse sec A r Hypotenuse x Side Adjacent tan A y Side Opposite x Side Adjacent cot A x Side Adjacent y Side Opposite SohCahToa Cofunction Identities For any acute angle A, sin A cos 90o A csc A sec 90 o A tan A cot 90 o A cos A sin 90o A sec A csc 90 o A cot A tan 90 o A Functions Values of Special Angles θ sin θ cos θ tan θ o 1 30 3 3 2 o 45 60o 2 2 3 2 2 2 2 1 2 cot θ sec θ 3 2 3 3 2 3 1 1 3 3 3 2 csc θ 2 2 2 3 3 Section 2 – 2 Trigonometric Functions of Non – Acute Angles Reference Angle – the positive acute angle made by the terminal side of Angle θ and the x – axis. Note: The reference angle must always be found from the x – axis! Finding Trigonometric Function Values for Any Nonquadrantal Angle Step 1 – If 360o , or if 0o , find a coterminal angle by adding or subtracting 360o as many times as needed to an angle 0o 360o Step 2 – Find the reference angle ' Step 3 – Find the necessary values of the trigonometric functions for the reference angle ' Step 4 – Determine if the correct signs for the values found in Step 3. Use the table in your notes or remember that All Students Take Calculus Section 2 – 3 Calculator Finding Trigonometric Function Values Using a Degree Mode Vs. Radian Mode To find the sec, csc, and cot. 1. Find the sin, cos, or tan of the desired angle 2. Use the x -1 button to find its reciprocal To find θ, the angle, use the sin -1, cos -1, or tan -1 buttons To find the csc -1, sec -1 or the cot -1, do the following, 1. Call up the reciprocal of the function you need. 2. Then take 1 value 3. It should look like this on you calculator sin 1 3.1245 1 Section 2 – 4 Solving Right Triangles Exact Number – a number that represents the result of counting, or a number that results from theoretical work and is not the result of a measurement Solving a Triangle – find the measures of all the angles and sides of the triangle. Angle of Elevation – the angle measured from the horizontal up Angle of Depression – the angle measured from the horizontal down Solving an Applied Trigonometry Problem Step 1 – Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. Step 2 – Use the sketch to write up an equation relating the given quantities to the variable. Step 3 – Solve the equation, and check that your answer makes sense. Section 2 – 5 Further Applications of Right Triangles Bearing – used in navigation. Two methods are used to solve problems involving bearings. Method 1 – when a single number is given as a bearing, it is understood that the bearing is measured in a clockwise direction from due north. Method 2 – when a bearing is given with a direction, then angle, then direction, (N 45o E) the bearing is measured from the first direction (Due North) in the rotation of the second direction East.