Maths Methods Trigonometric equations K McMullen 2012

The Unit Circle To help us understand the unit circle let’s first look at the right-angled triangle with a hypotenuse of 1 unit in length. The reason why it is 1 unit is because we consider the circumference of the circle to be 2π (circumference of a circle is 2πr, therefore r=1 which is the hypotenuse) this will be explained in more detail later. K McMullen 2012

The Unit Circle Using the cosine ratio in the triangle: cos ϑ =a/1 a=cos ϑ Using the sine ratio in the triangle: sin ϑ =b/1 b=sin ϑ This means that in a unit circle the horizontal length in the measure of the cosine of the angle and the vertical length is a measure of the sine of the angle. K McMullen 2012

The Unit Circle When this triangle is places inside a circle, it can be used to find the trig ratios shown previously. The value of cos ϑ can be read off the x-axis. The value of sin ϑ can be read off the y-axis. The value of tan ϑ can be read off a vertical tangent line drawn on the right side of the unit circle. A copy of the unit circle is in the next slide K McMullen 2012

The Unit Circle K McMullen 2012

Converting between radians and degrees Angles are measured in degrees or radians. To define a radian we can use a circle which has a radius of one unit. This circle is called the unit circle. When working with degrees we know that one revolution of a circle is 360°. When working with radians one revolution is 2π (this is because we are working with a circle with a radius of 1). Therefore: 360°=2π 180°=π K McMullen 2012

Converting between radians and degrees The radius of the circle can be any length and can still be regarded as a unit. As long as the arc is the same length as the radius, the angle will always measure one radian. An angle of 1 degree can be denoted as 1°. A radian angle can be denoted as 1 c but we usually leave off the radian sign. Therefore, 1 c = 180°π 1°= π180 K McMullen 2012

Converting between radians and degrees Always make sure your calculator is in radian mode when working with radians and degree mode when working with degrees. When working with radians and the unit circle we are no longer referring to North, East, South and West like we would with a compass. With the unit circle we use a set of axes (the Cartesian plane) with the x-axis as the horizontal and the y-axis as the vertical. Remember that cos ϑ =x and sin ϑ =y. K McMullen 2012

Converting between radians and degrees Below is a copy of the unit circle. You need to familiarize yourself with the values from this unit circle so make sure you remember the table of exact values that we did previously (it’s easy to recalculate these if needed by simply redrawing the two triangles). K McMullen 2012

Exact Values Using the equilateral triangle (of side length 2 units), the following exact values can be found: (look at page 271 to get the exact values- these go to the right of each ‘=‘ sign) sin 30°= sin π/6= sin 60°= sin π/3= cos 30°= cos π/6= cos 60°= cos π/3= tan 30°= tan π/6= tan 60°= tan π/3= K McMullen 2012

Exact Values Using the right isosceles triangle with two sides of length 1 unit, the following exact values can be found: sin 45°= sin π/4= cos 45°= cos π/4= tan 45°= tan π/4= K McMullen 2012

Symmetry Formulae The unit circle is symmetrical so that the magnitude of sine, cosine and tangent at the angles shown are the same in each quadrant but the sign varies. We’ll go over this in more detail in class K McMullen 2012

Trigonometric Identities When a right-angled triangle is placed in the first quadrant of a unit circle, the horizontal side has the length of cos ϑ and the vertical side has the length of sin ϑ. Therefore, using the tan ratio (tan ϑ =opposite/adjacent): tan ϑ = sin ϑ /cos ϑ Using Pythagoras’ theorem (a 2 +b 2 =c 2 ): (sin ϑ ) 2 +(cos ϑ ) 2 =1 2 sin 2 ϑ +cos 2 ϑ =1 K McMullen 2012

Complementary Angles Complementary angles add to 90° or π/2 radians. Therefore, 30° and 60° are complementary angles. In other words π/6 and π/3 are complementary angles, and θ and π/2- θ are also complementary angles. The sine of an angle is equal to the cosine of its complement. Therefore, sin 60°= cos (30°). We say that sine and cosine are complementary functions. The complement of the tangent of an angle is the cotangent or cot- that is, tangent and cotangent are complementary functions (as well as reciprocal functions). K McMullen 2012