Introduction to Trig Ratios and to Identity

Trig Ratios

\sin \theta = \dfrac{opp}{hyp}

\cos \theta = \dfrac{adj}{hyp}

\tan \theta = \dfrac{opp}{adj}

Reciprocal Identities

\csc \theta = \dfrac{1}{\sin \theta}

\sec \theta = \dfrac{1}{\cos \theta}

\cot \theta = \dfrac{1}{\tan \theta}

Pythagorean Identities

\sin^2 \theta + \cos^2 \theta = 1

Simplifying \cot x \sin x

ex Simplify cot \theta \times \sin \theta.

= \dfrac{\cos \theta}{\sin \theta} \times \sin \theta

= \cos \theta

Simplifying Trig Expression 1+\tan^2x

ex 1+\tan^2x

= 1+ (\dfrac{\sin \theta}{\cos \theta})^2

= 1 + \dfrac{\sin^2 \theta}{\cos^2 \theta}

= \dfrac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}

= \dfrac{1}{\cos^2 \theta}

= \sec^2 \theta

Simplifying Trig Expression \dfrac{1 + \sin \theta}{\cos^2 \theta}

ex Simplify \dfrac{1 + \sin \theta}{\cos^2 \theta}

= \dfrac{1 + \sin \theta}{1 - \sin^2 \theta}

= \dfrac{1 + \sin \theta}{(1 + \sin \theta)(1 - \sin \theta)}

= \dfrac{1}{1 - \sin \theta}

Simplifying Trig Expression

Introduction to Proving Identities

LS = RS

Prove that LS is the same as RS.

ex \dfrac{1}{\cos \theta} + \tan \theta = \dfrac{1 + \sin \theta}{\cos \theta}

Proving Identity with Relative Angles

Proving Identity ex2

ex Prove 2 \cos^2 \theta + \sin^2 \theta - 1 = \cos^2 \theta

Proving Identity ex3

ex Prove \tan \theta + \dfrac{1}{\tan \theta} = \dfrac{1}{\sin \theta \cos \theta}

For the proof of \sin^2B + \cos^2B = 1 using the triangle:

Fill in the missing step to prove.

\displaystyle \begin{array}{lllll} & LS &= (\frac{y}{r})^2 + (\frac{x}{r})^2\\ & &= \frac{y^2}{r^2} + \frac{x^2}{r^2}\\ & &= \frac{y^2+ x^2}{r^2} \\ & &= ......... \\ & &=1 \end{array}

Graph the relation y = \sin^2x + \cos^2x .

Prove the identity:

\sin \theta = \cos \theta \tan \theta

Which of the following could be a step going from right to left?

Prove the identity:

\displaystyle \csc \theta = \sec \theta \cot \theta

Prove the identity:

\displaystyle \cos \theta = \sin \theta \cot \theta

Which of the following could be a step going from right side to left?

Prove the identity:

\displaystyle \sec \theta = \csc \theta \tan \theta

Which of the following could be a step going from right side to left?

Prove the identity:

\displaystyle 1 +\csc \theta = \csc \theta(1 + \sin \theta)

Prove the identity:

\displaystyle \cot \theta \sin \theta \sec \theta = 1

Which of the following could be a step going from left side to right?

Prove the identity:

\displaystyle \cos \theta(\sec \theta - 1) = 1 -\cos \theta

Which of the following could be a step going from left side to right?

Prove the identity:

\displaystyle 1 + \sin \theta = \sin \theta(1 + \csc \theta)

Prove that \displaystyle 1 - \sin^2\theta = \sin \theta \cos \theta \cot \theta

Which of the following could be a step going from right side to left?

Prove that \displaystyle \csc^2\theta = \cot^2\theta + 1

Which of the following could be a step going from left side to right?

Prove that \displaystyle \frac{\cos \theta}{1 + \sin \theta} = \frac{1 - \sin \theta}{\cos \theta}

Start the proof by adding the rational expression on the left for \displaystyle \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta}= \frac{2}{\cos \theta}.

Show your step and simply.

Prove that \displaystyle \csc^2\theta \cos^2\theta = \csc^2\theta - 1.

Prove that \displaystyle \tan \theta +\cot \theta = \frac{\sec \theta}{\sin \theta}.

Which of the following could be a step going from left side to right?

Prove that \displaystyle \cot^2 \theta(1 +\tan^2\theta) =\csc^2\theta.

Which of the following could be a step going from left side to right?

You are on the last leg of the orienteering course. Determine the direction and distance from the information given. Draw the leg on your map. Label all angles and distances.

Direction: East of south

Determine the two angles between 0° and 360^o such that \frac{\csc \theta}{\sec \theta} = \cot \theta. Add their degree measures and divide by 9. Use this angle.

Hint Use identities to simplify each side of the equation first.

Distance: The result of evaluating

\displaystyle 20(\sec^2\theta \sin^2\theta + \sec^2\theta \cos^2\theta -\tan^2\theta\sin^2\theta - \tan^2\theta \cos^2\theta) , founded to the nearest metre if necessary.

Draw a right angle. Using the vertex as the centre, draw a quarter-circle that intersects the two arms of the angle. Select any point A on the quarter-circle, other than a point on one of the arms. Draw a tangent line to the quarter-circle at A such that the line intersects one arm at point B and the other arm at point C. Label \angle AOC as 6. Show that the measure of BA divided by the radius of the quarter-circle equals the cotangent of \angle \theta.

A student writes the following proof for the identity \cos \theta = \sin \theta \cot \theta. Critique it for form, and rewrite it in proper form.

Consider the equation \displaystyle \tan^2\theta -\sin^2\theta = \sin^2\theta\tan^2\theta.

a) Which of the basic identities will you use first to simplify the left side?

b) Which is the one step before simplifying to the right side?

If \sin^2\theta + \sin^22\theta+ \sin^23\theta = 1, what does \cos^23\theta + \cos^22\theta + \cos^2\theta equal?

Given \sin \theta\cot \theta = \frac{\sqrt{3}}{2}, find the value of \sin\theta .

Given \cos \theta = 3\sin \theta, a possible value for \cos^2\theta is

\displaystyle \begin{array}{cccccc} &(A)&0.25 &(B)& 0.7071 &(C) & 0.5 &(D) & 0.9 &(E) & 1 \\ \end{array}