This section looks at Sin, Cos and Tan within the field of trigonometry. A right-angled triangle is a triangle in which one of the angles is a right-angle. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The adjacent side is the side which is between the angle in question and the right angle. The opposite side is opposite the angle in question.  In any right angled triangle, for any angle: The sine of the angle = the length of the opposite side The cosine of the angle = the length of the adjacent side The tangent of the angle = the length of the opposite side So in shorthand notation: sin = o/h cos = a/h tan = o/a Often remembered by: soh cah toa Example Find the length of side x in the diagram below: The angle is 60 degrees. We are given the hypotenuse and need to find the adjacent side. This formula which connects these three is: cos(angle) = adjacent / hypotenuse therefore, cos60 = x / 13 therefore, x = 13 × cos60 = 6.5 therefore the length of side x is 6.5cm. This video will explain how the formulas work. The Graphs of Sin, Cos and Tan - (HIGHER TIER)
The following graphs show the value of sinø, cosø and tanø against ø (ø represents an angle). From the sin graph we can see that sinø = 0 when ø = 0 degrees, 180 degrees and 360 degrees. Note that the graph of tan has asymptotes (lines which the graph gets close to, but never crosses). These are the red lines (they aren't actually part of the graph). Also notice that the graphs of sin, cos and tan are periodic. This means that they repeat themselves. Therefore sin(ø) = sin(360 + ø), for example. Notice also the symmetry of the graphs. For example, cos is symmetrical in the y-axis, which means that cosø = cos(-ø). So, for example, cos(30) = cos(-30). For more information on trigonometry click here
Hypotenuse, Adjacent and Opposite Sides. In the following right triangle PQR,
Note: The adjacent and the opposite sides depend on the angle θ. For complementary angle of θ, the labels of the 2 sides are reversed. Example: Identify the hypotenuse, adjacent side and opposite side in the following triangle: a) for angle x b) for angle y Solution: a) For angle x: AB is the hypotenuse, AC is the adjacent side , and BC is the opposite side. b) For angle y: AB is the hypotenuse, BC is the adjacent side , and AC is the opposite side. SOH-CAH-TOAThe following diagram show how to use SOHCAHTOA for a right triangle. How to identify the Opposite Sides, Adjacent Sides and Hypotenuse of a Right Triangle? Definition of Cos, Sin, Tan, Csc, Sec, Cot for the right triangle sin x = opposite/hypotenuse cos x = adjacent/hypotenuse tan x = opposite/adjacent csc x = 1/sin x = hypotenuse/opposite sec x = 1/cos x = hypotenuse/adjacent cot x = 1/tan x = adjacent/opposite
Using the Sine Formula (the SOH formula) The first part of this video will explain the difference between the hypotenuse, adjacent and opposite sides of a right triangle. Then it shows how to use the sine formula (the SOH formula). Sine = Opposite over the Hypotenuse
Using the Cosine Formula (the CAH formula) Cosine = Adjacent over Hypotenuse
Using the Tangent Formula (the TOA formula) Tangent = Opposite over Adjacent
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc. RelatedTriangle Calculator | Pythagorean Theorem Calculator Right triangleA right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry. In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c, as shown below. In this calculator, the Greek symbols α (alpha) and β (beta) are used for the unknown angle measures. h refers to the altitude of the triangle, which is the length from the vertex of the right angle of the triangle to the hypotenuse of the triangle. The altitude divides the original triangle into two smaller, similar triangles that are also similar to the original triangle. If all three sides of a right triangle have lengths that are integers, it is known as a Pythagorean triangle. In a triangle of this type, the lengths of the three sides are collectively known as a Pythagorean triple. Examples include: 3, 4, 5; 5, 12, 13; 8, 15, 17, etc. Area and perimeter of a right triangle are calculated in the same way as any other triangle. The perimeter is the sum of the three sides of the triangle and the area can be determined using the following equation: Special Right Triangles30°-60°-90° triangle: The 30°-60°-90° refers to the angle measurements in degrees of this type of special right triangle. In this type of right triangle, the sides corresponding to the angles 30°-60°-90° follow a ratio of 1:√3:2. Thus, in this type of triangle, if the length of one side and the side's corresponding angle is known, the length of the other sides can be determined using the above ratio. For example, given that the side corresponding to the 60° angle is 5, let a be the length of the side corresponding to the 30° angle, b be the length of the 60° side, and c be the length of the 90° side.: Angles: 30°: 60°: 90° Ratio of sides: 1:√3:2 Side lengths: a:5:c Then using the known ratios of the sides of this special type of triangle: As can be seen from the above, knowing just one side of a 30°-60°-90° triangle enables you to determine the length of any of the other sides relatively easily. This type of triangle can be used to evaluate trigonometric functions for multiples of π/6. 45°-45°-90° triangle: The 45°-45°-90° triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45°-45°-90°, follow a ratio of 1:1:√2. Like the 30°-60°-90° triangle, knowing one side length allows you to determine the lengths of the other sides of a 45°-45°-90° triangle. Angles: 45°: 45°: 90° Ratio of sides: 1:1:√2 Side lengths: a:a:c Given c= 5: 45°-45°-90° triangles can be used to evaluate trigonometric functions for multiples of π/4. |
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