Trigonometric identities Trigonometric formulae

Angles Trigonometric ratios

Angles Reciprocal ratios

Right Triangle Trig Definitions B c a A C b sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a

Angles Pythagoras’ theorem The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides

Angles Special triangles Right-angled isosceles

Angles Special triangles Half equilateral

Special Right Triangles 30° 45° 2 1 60° 45° 1 1

Angles Special triangles Half equilateral

Trigonometric identities The fundamental identity The fundamental trigonometric identity is derived from Pythagoras’ theorem

Trigonometric identities Two more identities Dividing the fundamental identity by cos2

Trigonometric identities Two more identities Dividing the fundamental identity by sin2

Trigonometric formulas Sums and differences of angles Double angles Sums and differences of ratios Products of ratios

Trigonometric formulas Sums and differences of angles

Trigonometric formulas Double angles

Trigonometric formulas Sums and differences of ratios

Trigonometric formulas Products of ratios

Remember an identity is an equation that is true for all defined values of a variable. We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have.

RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES EVEN-ODD IDENTITIES

Even and Odd Functions An odd function is one where f(-x) = -f(x) and an even function is one where f(-x) = f(x).

Let's sub in here using reciprocal identity Establish the following identity: Let's sub in here using reciprocal identity We are done! We've shown the LHS equals the RHS We often use the Pythagorean Identities solved for either sin2 or cos2. sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our left-hand side so we can substitute. In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

Let's sub in here using reciprocal identity and quotient identity Establish the following identity: Let's sub in here using reciprocal identity and quotient identity We worked on LHS and then RHS but never moved things across the = sign FOIL denominator combine fractions Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom

Get common denominators Hints for Establishing Identities Get common denominators If you have squared functions look for Pythagorean Identities Work on the more complex side first If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match!

See what you get

Etc.

Change everything on both sides to Suggestions Start with the more complicated side Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier) Try algebra: factor, multiply, add, simplify, split up fractions If you’re really stuck make sure to: Change everything on both sides to sine and cosine.

All Students Take Calculus. Quad I Quad II cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV Quad III

Reference Angles Quad I Quad II θ’ = 180° – θ θ’ = θ θ’ = π – θ θ’ = θ + 180° θ’ = 360° – θ θ’ = θ + π θ’ = 2π – θ Quad III Quad IV

Trigonometric Identities Summation & Difference Formulas

Trigonometric Identities Double Angle Formulas

Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.

Work with identities to simplify an expression with the appropriate trigonometric substitutions. For example, if x = 3 tan , then

Expressing One Function in Terms of Another Example Express cos x in terms of tan x. Solution Since sec x is related to both tan x and cos x by identities, start with tan² x + 1 = sec² x. Choose + or – sign, depending on the quadrant of x.

Rewriting an Expression in Terms of Sine and Cosine Example Write tan  + cot  in terms of sin  and cos . Solution

9.1 Verifying Identities Learn the fundamental identities. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. It is often helpful to express all functions in terms of sine and cosine and then simplify the result. Usually, any factoring or indicated algebraic operations should be performed. For example, As you select substitutions, keep in mind the side you are not changing, because it represents your goal. If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

Verifying an Identity ( Working with One Side) Example Verify that the following equation is an identity. cot x + 1 = csc x(cos x + sin x) Analytic Solution Since the side on the right is more complicated, we work with it. Original identity Distributive property The given equation is an identity because the left side equals the right side.

Verifying an Identity Example Verify that the following equation is an identity. Solution

Verifying an Identity ( Working with Both Sides) Example Verify that the following equation is an identity. Solution

Verifying an Identity ( Working with Both Sides) Now work on the right side of the original equation. We have shown that

Using the unit circle r

Draw the graph of On the same axis: Add the graphs together to make the blue line shown below. The graph of

Two more identities starting with Divide by Divide by

Prove the following: use the identity

Try these with no hints

The unit circle

Complete the radian row of this table using exact numbers 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360

Radians should be: 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360

The Unit Circle 1 Consider the circle shown opposite. y Make a right angled triangle. An expression for x: (x,y) 1 y x x An expression for y: And now use tan:

The Unit Circle - the second quadrant Make an expression for x: 1 Make an expression for y: y -x Make an expression for tan: What about the third and fourth quadrants?

Trig. ratios in the 1st quadrant (1) 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table 2 1

Trig. ratios in the 1st quadrant (2) 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table 1 1

Trig. ratios in the 1st quadrant (3) 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table 2 1

Trig. ratios in the 2nd quadrant 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table 2 -1 Use your answers from previous slides to fill in all the angles trig. values in the second quadrant.

Trig. ratios in the 3rd quadrant 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table -1 -1 Use your answers from previous slides to fill in all the angles trig. values in the third quadrant.

Trig. ratios in the 4th quadrant 1. Find the missing side, giving your answer as a surd. 2. Find an expression for sin . 3. Find an expression for cos . 4. Find an expression for tan . 5. Find the value of the angle, using any of 2-4. 6. Fill in the appropriate column of your table -1 2 Use your answers from previous slides to fill in all the angles trig. values in the second quadrant.

The completed table 30 45 60 90 120 135 150 180 1 -1 - 210 225 240 270 300 315 330 360 -1 1 -