A baseball fan is sitting directly behind home plate in the last row of the upper deck of U.S. Cellular Field in Chicago.  The angle of depression to home plate is 29°54’, and the angle of depression to the pitcher’s mound is 24°12’.  In major league baseball, the distance between home plate and the mound is 60.5 feet.  How far is the fan from home plate?  To solve this scenario, we must look to a new law of Mathematics, the Law of Sines, to be used to solve triangles that are 

This new law can be used to solve any type of triangle.  Here is a proof that will determine the common law that can be applied.  Consider this image:

 with the diameter as segment DB.  The length of the diameter is defined as 2

, or twice the radius.  Segment AD is drawn to connect the intercept of the diameter with the edge of the circle and angle A of the inscribed triangle.  Given corollary: Two angles inscribed in a circle that intercept the same arc are congruent.  Therefore, ∠D ≊ ∠C since they both intersect arc AB.  So, sin D = sin C.  ∠DAB is inscribed in a semicircle, so it is a right angle as per that corollary.  In the current setup, sin D = c / 2r, sin D = sin C, and so sin C = c / 2r, or c / sin C = 2r.

We can use the same process with angles A and C, drawing a diameter from there, and end up with b / sin B = 2r and a / sin A = 2r, leading us to state that the ratio of the length of any side of a triangle to the sine of the opposite angle is a constant for any given triangle.  Ah, the Law of Sines:

Draw the triangle above in your Math journal.  Now, let’s solve ΔABC if A = 33°, B = 105° and 

 = 37.9.  🔑What is the first measurement that you can find?

 is given.  You can now use the Law of Sines to find 

.  You may use calculators from here on out.  Record the value of your solved triangle.

Refer to the baseball application from earlier in the lesson.  🔑Draw a diagram of the information given between two parallel horizontals.  How do the two given angles of depression create a triangle?  What is the measure of the interior angle for the top of this triangle, at the spectator’s seat?

You should now have two angle measures and one adjacent side representing the distance (

) from home plate and the mound.  🔑Use the Law of Sines to find the distance from home plate to the baseball fan in the top row.

 ΔABC with corresponding sides in your Math journal, this time containing an altitude (

.  Given: The area of the triangle will be expressed as K.  Then, K = ½

 sin A.  If you substitute this value into the original area formula, then K = ½

 sin A.  🔑Now, using an altitude drawn from A and from C, develop two other formulas using these for the area of ΔABC.  These three formulas will apply for any triangle given any mix of information of sides and angle measurements.

The same style of process can be applied if you have two angles and one side in a Pythagorean-esque relationship.  Follow this: By the Law of Sines, 

 sin B) / sin C.  Substitute that in for 

                                               K = ½ c

🔑That formula applies when the given side is 

.  Find the formulas when the given sides are 

, respectively.  👩‍🔬Note: In these situations, you will always have all three angle measurements, thanks to a little arithmetic👩‍🔬.

🔑Solve the prompts below for ΔABC.  Use a calculator when necessary (often).  Round to the nearest tenth on lengthy decimal solutions.  Draw a diagram for the word problems if needed.

Students will apply trigonometric knowledge to triangles without a right angle, and how to solve them.

I Saw the Sine!