Table of Contents 1. Angles and their Measures

Angles and their Measures Essential question – What is the vocabulary we will need for trigonometry?

Make a table TermDefinitionPicture

Trigonometry vocabulary Initial side – start side of angle Terminal side – end side of angle Standard position – An angle whose initial side is on the positive x-axis

Positive angles An angle in standard position that rotates counterclockwise

Negative angles An angle in standard position that rotates clockwise

Coterminal Angles Angles that have the same terminal side

Quadrants Quadrant III Quadrant I Quadrant II Quadrant IV

Angles of the axes

Variables you will see for angle measures

Radians Angle measures can also be expressed in radians A radian is the ratio of the length of an arc to its radius Radians are expressed in terms of = 180 o To change from degrees to radians, multiply by and reduce. To change from radians to degrees, multiply by

Radians continued Radians can take 2 forms – an exact answer and an approximate decimal answer The exact answer has a in it and it is the usual way to see radians To find an exact answer with your calculator, do not put the in the calculator, only write it in the answer However, radians can also be written as a decimal without the

Angles of the axes

Examples Change from degrees to radians Change from radians to degrees

Coterminal angles You add or subtract multiples of 360 o (or 2π) to find coterminal angles Find 2 coterminal angles (one positive and one negative) for 35 o Find 2 coterminal angles (one positive and one negative) for -23 o Find 2 coterminal angles (one positive and one negative) for 740 o

Examples for radians Find a positive and negative coterminal angle

What quadrant is it in? To find out what quadrant an angle is in –Make a negative angle positive by adding 360 o or 2π (may need to do multiple times) –If angle is bigger than 360 o or 2π, make it smaller by subtracting 360 o or 2π (may need to do multiple times) –Figure out what quadrant it is in based on angles of axes (from yesterday) –If the question asks you to sketch the angle, draw the terminal side in the right quadrant go in either positive or negative direction based on original problem if you have added or subtracted 360 o or 2π, you need to go around multiple times.

What quadrant is it in (and sketch)?

What quadrant is it in? (radians) Follow steps to make small positive angle Put fraction in calculator (without the π) If answer is < 0.5, it is in 1 st quadrant If answer is between 0.5 and 1, it is in 2 nd quadrant If answer is between 1 and 1.5, it is in 3 rd quadrant If answer is between 1.5 and 2, it is in 4 th quadrant

Examples – which quadrant? (radians) (and sketch)

Reference Angles A reference angle is the acute angle that an angle makes with the x-axis

Finding Reference Angles Follow steps to make small positive angle Find out which quadrant it is in In the 1 st quadrant, the reference angle is the SAME as the angle itself In the 2 nd quadrant subtract the angle from 180 o or π In the 3 rd quadrant subtract 180 o or π from the angle In the 4 th quadrant subtract the angle from 360 o or 2π

Examples Find the reference angle for the following angles. 37 o 7π/4 -2π/ o 17π/7 820 o

Assessment 321 –Write 3 new things you learned –Write 2 vocabulary words with their meaning –Write 1 thing you don’t understand