TRIGONOMETRY – Functions 2 In the previous section we were finding sin, cos, and tan for given right triangles. We also learned how to convert an angle into its decimal equivalent and vice versa. In this section, we will develop the unit circle, and find unknown sides and angles in given right triangles. +y θ - x +x - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. +y 1 θ - x +x - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y 1 30⁰ 60⁰ - x +x - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y Recall from Geometry, the side opposite the 30⁰ angle is ½ the hypotenuse. 1 30⁰ 60⁰ - x +x - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y   1 30⁰ 60⁰ - x +x   - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y   1 30⁰ 60⁰ - x +x   - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y   1 30⁰   60⁰ - x +x   - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y So now for the given angle of 60⁰ we have : 1     60⁰ - x +x       - y

TRIGONOMETRY – Functions 2 Let’s go back to our original example where the hypotenuse of the drawn right triangle = 1. If we let the angle = 60⁰, we create a 30 – 60 – 90 right triangle. +y This creates the coordinate point :   So now for the given angle of 60⁰ we have : 1     60⁰ - x +x       - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). +y   1 - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE +y   1 - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y The unit circle is a circle with radius = 1. It will help us with finding the coordinate values of specific angles using cos, sin & tan.   1 - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y Let the given angle = 45⁰   1 45⁰ - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y Let the given angle = 45⁰ This creates a 45 – 45 – 90 right ∆   1 a 45⁰ - x +x a - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y Let the given angle = 45⁰ This creates a 45 – 45 – 90 right ∆   Using pythagorean theorem : 1   a   45⁰ - x +x a       - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y Let the given angle = 45⁰ This creates a 45 – 45 – 90 right ∆   Using pythagorean theorem : 1       45⁰ - x +x           - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). We will use this format to create the UNIT CIRCLE We will do this for specific angles in the first quadrant, and the remaining angles will be left as an exercise for you to complete. +y Let the given angle = 45⁰ This creates a 45 – 45 – 90 right ∆ So the ( cos , sin ) coordinate is :   Using pythagorean theorem : 1       45⁰ - x +x           - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). Now let’s get some easy ones… +y - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). Now let’s get some easy ones… +y Let the given angle = 0⁰   - x +x 1 - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). Now let’s get some easy ones… +y Let the given angle = 0⁰     - x +x 1 - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). Now let’s get some easy ones… +y Let the given angle = 0⁰     1 Let the given angle = 90⁰ - x +x - y

TRIGONOMETRY – Functions 2 When describing angles, we will use a coordinate point that will always have the form ( cos , sin ). Now let’s get some easy ones… +y Let the given angle = 0⁰     1 Let the given angle = 90⁰ - x +x   - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. +y 1 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. But we still have a 30 – 60 – 90 right triangle . +y   1 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. But we still have a 30 – 60 – 90 right triangle . The angle in reference to the negative x – axis is till 60 degrees. +y   1 60⁰ 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. But we still have a 30 – 60 – 90 right triangle . The angle in reference to the negative x – axis is till 60 degrees. So the ( cos , sin ) coordinate is the same, it just takes on the quadrant signs. +y   1 60⁰ 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. But we still have a 30 – 60 – 90 right triangle . The angle in reference to the negative x – axis is till 60 degrees. So the ( cos , sin ) coordinate is the same, it just takes on the quadrant signs. +y   These similar angles occur in each quadrant and have the same ( cos , sin ) values. The only thing that changes is the sign. 1 60⁰ 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 If we take the 30 – 60 – 90 right triangle we started with and mirror it around the y – axis, the 60 degree angle becomes 120 degrees. But we still have a 30 – 60 – 90 right triangle . The angle in reference to the negative x – axis is till 60 degrees. So the ( cos , sin ) coordinate is the same, it just takes on the quadrant signs. +y   These similar angles occur in each quadrant and have the same ( cos , sin ) values. The only thing that changes is the sign. You will complete the unit circle chart in the practice problems. 1 60⁰ 120⁰ - x +x - y

TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle

a b TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. a 12 52⁰ b

a b TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. 1. Label your sides a hyp opp 12 52⁰ b adj

a b TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. Label your sides Pick a missing side to find a hyp opp 12 52⁰ b adj

a b TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. Label your sides Pick a missing side to find Which function contains your given information and unknown ? a hyp opp 12   52⁰ b adj

a b TRIGONOMETRY – Functions 2 Trigonometry can be used to find missing sides and angles of right triangles. ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown a hyp opp 12   52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.   a hyp   opp 12 52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.   a hyp   opp 12   Change your degrees to decimal equivalent 52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.   a hyp   opp 12     52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.     a hyp   opp 12     52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.       a hyp   opp 12     52⁰ b adj

a b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.       a hyp   opp 12     Round to 2 decimal places   52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle. To find ‘b” you could use pythagorean theorem, but to show another example I will use trig. hyp opp 12   52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.     hyp opp 12   52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.     hyp opp 12     Change your degrees to decimal equivalent 52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.     hyp opp 12       52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.       hyp opp 12       52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.         hyp opp 12       52⁰ b adj

b TRIGONOMETRY – Functions 2 Label your sides Pick a missing side to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 1 : Find the missing sides of the given right triangle.         hyp opp 12         52⁰ b adj  

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.   8.1 5.6  

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.   Sides for   hyp 8.1 opp 5.6   adj

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for   hyp 8.1 opp 5.6   adj

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for     hyp 8.1 opp 5.6     adj

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for     hyp 8.1 opp 5.6         adj

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.   Sides for   hyp 8.1 adj 5.6   opp

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for   hyp 8.1 adj 5.6   opp

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for     hyp 8.1 adj 5.6     opp

TRIGONOMETRY – Functions 2 Label your sides Pick an angle to find Which function contains your given information and unknown ? Plug in your given info and solve for the unknown ONE of the following conditions must be met : a) you know two sides, OR b) you know 1 side and 1 angle EXAMPLE # 2 : Find the missing angles in the given right triangle.     Sides for     hyp 8.1 adj 5.6         opp

TRIGONOMETRY – Functions 2 EXAMPLE # 3 : Find the angle created by the coordinate ( -2 , - 7 ) +y - x +x - y

TRIGONOMETRY – Functions 2 EXAMPLE # 3 : Find the angle created by the coordinate ( -2 , - 7 ) +y Labeling the sides and using tangent :   -2 - x +x     -7   - y

TRIGONOMETRY – Functions 2 EXAMPLE # 3 : Find the angle created by the coordinate ( -2 , - 7 ) +y Labeling the sides and using tangent :   -2 - x +x     -7       - y

TRIGONOMETRY – Functions 2 EXAMPLE # 3 : Find the angle created by the coordinate ( -2 , - 7 ) But the coordinate is in Quadrant 3, so we must add 180⁰ to our answer to find the true measure of the angle. +y Labeling the sides and using tangent :   -2 - x +x     -7       - y