Geometry and trigonometry

angles triangles circles trig volume

The SAT loves their circles and triangles...

All of the required concepts have been covered in detail for AAHL and AIHL courses, but AASL and AISL classes may have some things to learn.

Trigonometric values

Please be familiar with the conversion between degrees and radians. As a reminder, the relation between radians and degrees is as follows:

Conversion between radians and degrees:

Here are good trigonometric ratios to remember. The angles on the top are in degrees and on the bottom in radians. N.D stands for not defined.

Unit Circle Summary.pdf

Worked example

This problem uses the concept of the unit circle and radian angle values.

Using the unit circle, we know that one revolution of the circle has a measure of 2π rad. Therefore, a simplification of the tangent value can be found by subtracting multiples of 2π.

We know π/2 < 2π/3 < π, therefore this is in quadrant 2. Since tan(π/3) = sqrt(3), the value of tan(2π/3) must equal -sqrt(3). Hence, answer choice A is correct.

A more detailed answer choice justification can be found here.

Circle relation

Below is the general formula for the circle relation on a graph, where (h,k) is the centre of the circle and r is the radius.

Worked example

Using the above formula for circles, it can be seen that h=3 and r=√2. Inserting those values gives.(x-3)^2+(y-k)^2=2.

Since the circle passes through the point (4,5). We can insert the values into the equation.

1+(5-k)^2=2

5-k=±1

k=4 or 6

Therefore, the possible coordinate of the circle is (3,4

Special triangles

This includes the properties of the 30-60-90 triangle and 45-45-90 triangle.

x represents the lengths of the triangle sides.

Worked example

This problem uses special angles.

From the diagram, the triangle is a right triangle. Using right-angle trigonometry, it can be seen that cos(QRS)= (10√5)/(5√5) = 1/2. Therefore, angle RQS = 60 degrees.

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