We learned about trigonometry by learning about the formula known as Pythagoras Theorem. This formula is used to find a missing length of any right triangle by using the known side lengths and actually using them in the equation. The equation a2 + b2 = c2 can also be moved around if you were to have a and c and needed to find b. To prove the Pythagorean Theorem, we were taught to look at a triangle divided into two smaller right triangles. We then can prove the theorem by taking apart the triangle into 3 similar parts and prove that a2 + b2 = c2 . To prove distance formula, we can use a2 + b2 = c2 but with different labels. This equation which is shown below can be used to derive the equation of a circle centered at the origin of a Cartesian coordinate plane aka the unit circle.
The circle we used had a radius of 1 which made everything a little more simple to learn from. All points of any circle that are the same length from a point are known as locus. We proved that all circles can be translated and dilated to be congruent. the equation x^2 + y^2 = r^2 which is mirrored to the pythagorean theorem and the distance formula can be used to find points on the unit circle. To find points on the unit circle with angles of 30 degrees, 45 degrees and 60 degrees we did something very similar and used the formula x^2 + y^2 = 1. When finding the point for 30 degrees we can reflect the triangle over the x-axis creating an equilateral triangle and solve because we will know y is now ½. We know this because of solving for 45. When solving for 45 degrees we know the x and y are equal which helps with solving. when solving for a triangle at 60 degrees you would do it very similar to solving for 30 by reflecting over the y-axis. we also learned about Law of Sines and the Law of Cosines. We did this by using the mount everest problem. This problem is where a group of people are trying to find the peak of mount everest with only two angles and one side length. The Law of Cosines is used when you know two side lengths and one angle. The Law of Sines is used when you have two angles and one side length. The other part of these equations that differentiates them from the pythagorean theorem is that they are used with non right triangles
The circle we used had a radius of 1 which made everything a little more simple to learn from. All points of any circle that are the same length from a point are known as locus. We proved that all circles can be translated and dilated to be congruent. the equation x^2 + y^2 = r^2 which is mirrored to the pythagorean theorem and the distance formula can be used to find points on the unit circle. To find points on the unit circle with angles of 30 degrees, 45 degrees and 60 degrees we did something very similar and used the formula x^2 + y^2 = 1. When finding the point for 30 degrees we can reflect the triangle over the x-axis creating an equilateral triangle and solve because we will know y is now ½. We know this because of solving for 45. When solving for 45 degrees we know the x and y are equal which helps with solving. when solving for a triangle at 60 degrees you would do it very similar to solving for 30 by reflecting over the y-axis. we also learned about Law of Sines and the Law of Cosines. We did this by using the mount everest problem. This problem is where a group of people are trying to find the peak of mount everest with only two angles and one side length. The Law of Cosines is used when you know two side lengths and one angle. The Law of Sines is used when you have two angles and one side length. The other part of these equations that differentiates them from the pythagorean theorem is that they are used with non right triangles