Learn to measure angles in radians. Radians are an alternate to degrees as a way of measuring angles. In 180o, there are pi, or approximately 3.142, radians. Radians are especially useful when it comes to investigating the properties of a circle, and are also used in physics in the study of waves and simple harmonic motion.
Move on to non-right triangles.. Because non-right triangles do not have a right angle (that’s kind of the definition), the three trigonometric ratios play a smaller role here (although they can also be used in some situations). Rather, two other rules become very important: The Sine Rule, and The Cosine Rule. The following articles explain these rules in detail.
Start with studying right-angled triangles. Right angled triangles are easy to study and will give you a good grasp of basic trigonometry and the three trigonometric ratios.
- Familiarize yourself with the three sides of a right-angled triangle.
- The hypotenuse is the side opposite the right angle. It is the biggest side of any right triangle.
- The two other sides are called the legs of the triangle. If you pick any angle in the triangle (besides the right angle), you will see that one leg is adjacent to the angle, and the other leg is opposite the angle.
- Familiarize yourself with the three trigonometric ratios, the base of trigonometry:
- The Sine of any angle is the ratio of the length of the side opposite it to the length of the hypotenuse.
- The Cosine of any angle is the ratio of the length of the side adjacent to it to the length of the hypotenuse.
- The Tangent of any angle is the ratio of the Sine of the angle to the Cosine of the angle. It is often also taken as the ratio of the opposite to the adjacent. The first definition is especially of help in solving trigonometric equations and proving identities while the second is sufficient for a basic study of trigonometry.
Brush up your basic mathematical skills. These include knowledge of algebra and algebraical manipulation as well as geometry.
- Practice algebraic manipulation. Algebraic manipulation is a very basic skill that is necessary to study any branch of mathematics.
- Learn to change the subject of any equation.
- Learn to solve linear and quadratic equations.
- Learn to solve simultaneous linear equations and linear/quadratic pairs of simultaneous equations.
- Learn basic geometry. Geometry is very closely related to trigonometry and plays a vital part in solving trigonometric problems.
- Learn the properties of a circle.
- Learn the properties of the interior and exterior angles of polygons including triangles.
- Learn the three different types of triangles i.e. isosceles, equilateral, and scalene.
Trigonometric functions on the unit circle
The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (x, y) is a point of the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle t from the positive x-axis, (where counterclockwise turning is positive), then
The equation x2 + y2 = 1 gives the relation
for any integer k.
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(x1,y1) on the unit circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ OQ. The result is a right triangle ΔOPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(−x1,0) and line segments RS OS. The result is a right triangle ΔORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π−t, R is at (cos(π−t),sin(π−t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (−x1,y1) is the same as (cos(π−t),sin(π−t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π−t) and −cos(t) = cos(π−t). It may be inferred in a similar manner that tan(π−t) = −tan(t), since tan(t) = y1/x1 and tan(π−t) = y1/(−x1). A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/sqrt(2).
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the Sum and Difference Formulas.
Forms of unit circle points
- exponential :
- trigonometric :
From Wikipedia, the free encyclopedia
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, “the” unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.
If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation
- x2 + y2 = 1.
Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not just those in the first quadrant.