The longest punt in NFL history was for 98 yards.  The punt was made by Steve O’Neal of the New York Jets, which, believe it or not, was when they were actually 

 back in 1969.  When a football is punted, the angle made by the initial path of the ball and the ground affects both the height and the distance the ball will travel.  If a football is punted from ground level, the maximum height it will reach is given by the formula: h = (v

 is the initial velocity, θ is the measure of the angle between the ground and the initial path of the ball, and g is the acceleration due to gravity (9.8 mps

).  Let’s look more closely at the trigonometric side of this proposal when given a trajectory angle is made between 0 and 90°.

Picture, if you will, a circle drawn about a regular coordinate system, centered at the origin, with a radius of 1.  This circle is symmetric with respect to the x-axis, the y-axis and the origin.

The radius forms the hypotenuse of a right triangle that can be inscribed, with legs having a length of 

 respectively.  Therefore, the radius meets the edge of this circle at the coordinate (

) and creates an angle, θ, relative to the x axis.  Leg 

) to the x-axis.  Now, according to the Pythagorean Theorem in this case, x

 = 1, and we can find values for sin θ and cos θ using the definitions of the trigonometric ratios.  So, sin θ = y/1 and cos θ = x/1, and can be redefined as the 

🔑 📓If sin θ = y/1 and cos θ = x/1, what would be the value of tan, csc, sec, and cot of angle θ?  Fill out the table below

Let’s use this information to look back at the Engage information.  Use the formula given, including gravity and the variables listed.  For this example, consider that the initial velocity of the ball is 28 mps

.  🔑📓Describe the maximum possible height of the ball, if the initial angle is between 0 and 90°, in your Math Journal or in the notebook on this page.

So, How Does This Concept Apply to Angle  Measurement?

angle in standard position, or, an angle whos terminal side does not lie on one of the axes of a coordinate plane, you can use the acute angle formed by the terminal side and the next closest quadrant axis in a rotation.  This acute angle is known as the 

.  Measuring the reference angle involves subtraction and the original angle.  Follow the reference “formulas” below when the terminal side of the angle is in a certain quadrant.  For ∠θ in standard position, the measure in degrees of its reference angle is:  

🔑What are four angles between 0 and 360° that have a reference angle of 20° ?

Use the “formulas” above and write your answer to the prompt in your Math Journal or in the notebook on this page.

When looking into reference angles, there are some very special right triangles to keep in mind.  The lengths of the sides of triangles with rational angle measures are equally rational.  These special angle relationships include triangles with 30°- 60°- 90° and 45°- 45°- 90° combinations. 

 So, if a right triangle made from the radius of 1 of the unit circle contains a reference angle with an internal acute angle of 60°, its adjacent leg would measure ½.  The following trigonometric ratios are good to keep in mind when working with these special triangles:

🔑📓Using what you know about the unit circle, identify the following information you can infer from a 135° angle.

You can also find the measure of an angle based solely on the coordinates of (

) made by the terminal side of a given angle.  If the coordinates lie inside or outside the unit circle radius of 1, the rules still apply to the trigonometric ratios already defined.  Instead of 1 as the value of the hypotenuse of the created right triangle, it can be defined as 

.   You can find the trigonometric values of any given angle with only the information of the intersection point (

🔑📓Solve the following prompts considering an angle θ in standard form and the terminal side's intersection point at (5 , -12).

🦄📓Solve the following prompts where the ∠θ has a terminal side that lies in Quadrant III and the sin θ = -(4 / 5).

Students will apply knowledge of right triangles to an investigation of the unit circle and its measurements.

Seekers of the Lost Arc