Grade 12 – Advanced Functions
- Angle X is defined as the length, a, the arc that extends the angle divided by the radius, r, of the circle: X = a/r.
- 2 Pi rad = 360 degrees or Pi rad = 180 degrees.
- To convert degrees to radians, multiply degree measure with Pi/180.
- To convert radian measure to degrees, multiply radian measure with 180/Pi.
Trigonometric ratios and special angles
- You can use a calculator to calculate trigonometric ratios for an angle expressed in radian measure by setting the angle mode to radians.
- You can determine the reciprocal trigonometric ratios for an angle expressed in radian measure by first calculating the primary trigonometric ratios then using the reciprocal key on the calculator.
- You can also use the unit circle and special triangles to determine exact values for the trigonometric ratios of the special angles 0, Pi/6, Pi/3. Pi/4, and Pi/2.
- You can use the unit circle along with the CAST rule to determine exact values for the trigonometric ratios of multiples of the special angles.
Equivalent Trigonometric Expressions
- You can use a right triangle to derive equivalent trigonometric expressions that form the cofunction identities, such as sinx = cos(Pi/2 – x).
- You can use the unit circle along with transformations to derive equivalent trigonometric expressions that form other trigonometric identities, such as cos(Pi/2 + X) = -sinx
- Given a trigonometric expression of a known angle, you can use the equivalent trigonometric expressions to evaluate trigonometric expressions of other angles.
- You can use graphing technology to demonstrate that two trigonometric expressions are equivalent. Some of which are known as the Co-Functions Identities.
Compound Angle Formula
- You can develop compound angles using algebra and the unit circle.
- Once you have developed one compound angle formula, you can develop others by applying equivalent trigonometric expressions.
- The compound angle, or addition and subtraction, formulas for sine and cosine are:
- sin(X+Y) = sinXcosY+cosXsinY
- sin(X-Y) = sinXcosY-cosXsinY
- cos(X+Y) = cosXcosY-sinXsinY
- cos(X-Y) = cosXcosY+sinXsinY
- tan(X+Y) = tanX+tanY / 1-tanXtanY
- tan(X-Y) = tanX-tanY / 1+tanXtanY
- These identities can also be made into more identites:
- cos2X = cos^2X – sin^2X
- = 1 – 2sin^2X
- = 2cos^2X-1
- sin2X = 2sinXcosX
- tan2X = 2tanX / 1-tan^2X
- cos2X = cos^2X – sin^2X
Prove Trigonometric Identities
- A Trigonometric identity is an equation which trigonometric expressions that is true for all angles in the domain of the expressions on both sides.
- One way to show that an equation is not an identity is to determine a counter example
- To prove that an equation is an identity, treat each side of the equation independently and transform the expression on one side into the exact form of the expression on the other side.
- The basic trigonometric identites are the Pythagorean identity, the quotient identity, the reciprocal identities, the compound angle formulas. You can use these identities to prove more complex identities.
- Trigonometric identities can be used to simplify solutions to problems that result in trigonometric expressions.