Summary
Highlights
Trigonometry is a branch of mathematics dealing with the relationship between sides and angles of a triangle, originating from Greek words meaning "measurement of triangles." It's used for measuring distances indirectly and focuses on angles and geometric ratios. In right triangle trigonometry, the hypotenuse is always opposite the right angle and is the longest side. The opposite and adjacent sides are defined relative to a specific acute angle, and their roles can interchange depending on the angle being considered.
The six trigonometric ratios are based on the relationships between the opposite, adjacent, and hypotenuse sides of a right triangle. The three primary ratios are: sine (opposite over hypotenuse), cosine (adjacent over hypotenuse), and tangent (opposite over adjacent). These can be remembered using the acronym SOH CAH TOA. The other three ratios are their reciprocals: cosecant (hypotenuse over opposite), secant (hypotenuse over adjacent), and cotangent (adjacent over opposite).
The video provides an example using triangle ABC with sides 3, 4, and 5. It shows how to calculate the six trigonometric ratios for angle A. For sine A, it's 3/5 (opposite/hypotenuse); for cosine A, it's 4/5 (adjacent/hypotenuse); and for tangent A, it's 3/4 (opposite/adjacent). The reciprocal ratios (cosecant, secant, cotangent) are then easily derived by inverting these fractions. The same process is then demonstrated for angle B, highlighting how the opposite and adjacent sides change depending on the reference angle.
Another example demonstrates finding the six trigonometric ratios for an angle theta when one side of the right triangle is unknown. The Pythagorean theorem (c² = a² + b²) is used to find the missing hypotenuse. With sides 6 and 8, the hypotenuse is calculated to be 10. Once all sides are known, the sine, cosine, and tangent are calculated and expressed in their lowest terms, followed by their reciprocal ratios.
This section presents an example where a leg of the right triangle is missing. Given a hypotenuse of 17 and one leg of 15, the Pythagorean theorem (x² = 17² - 15²) is used to find the missing leg, which turns out to be 8. The video then proceeds to calculate various trigonometric ratios for both angle X and angle Y (sine X, cosine X, cosecant Y, tangent Y, secant X), demonstrating the application of SOH CAH TOA and reciprocals.
An example involving a right triangle with sides 3 and sqrt(10) is given. The hypotenuse is found to be sqrt(19) using the Pythagorean theorem. When calculating sine D, cosine D, and tangent D, the results initially have square roots in the denominator. The video explains the process of rationalizing the denominator by multiplying the numerator and denominator by the radical to simplify the expression.
If sine theta is given as 5/13, this implies the opposite side is 5 and the hypotenuse is 13. The Pythagorean theorem helps find the missing adjacent side, which is 12. With all three sides known, the remaining five trigonometric ratios (cosine, tangent, cosecant, secant, cotangent) can be easily determined based on their definitions.
Given tangent theta as 5/(2*sqrt(6)), an unknown hypotenuse 'x' is determined using the Pythagorean theorem (x² = 5² + (2*sqrt(6))²), resulting in x=7. The task is to find cosine theta (adjacent/hypotenuse) and tangent alpha (opposite/adjacent for alpha), demonstrating the importance of correctly identifying sides relative to the specified angle.
The final section focuses on determining the correct trigonometric equation to find a missing side. Using an example where an angle (54 degrees), the hypotenuse (12), and the adjacent side (y) are known, the cosine function (cosine x = adjacent/hypotenuse) is chosen to set up the equation y = 12 * cosine 54 degrees. Similarly, to find the opposite side (x) with the same given, sine is used, resulting in x = 12 * sine 54 degrees. Another example showcases finding the adjacent side 'a' and hypotenuse 'c' when an angle (67 degrees) and the opposite side (10.6) are known, using tangent and sine respectively, including steps for algebraic manipulation to isolate the unknown variable.