How to Solve a Trigonometry Triangle Question Using Cosine and Sine Functions

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In summary, to find the type of triangle A, B, and C form based on the given equation, you can use the fact that sin^2 + cos^2 = 1 and the given equation is similar to this. By setting B = C, you can determine that two angles are the same. Then, by solving for sinA, you can find that A must be a right angle, leaving you with a right angled isosceles triangle.
  • #1
mercury
this is a question that was part of an mcq test.
and i did'nt have any clue as to how to begin!
finally i just tried plugging in values for A,B,C to see if they worked.
for ex. i put A=B=C=60 (degrees)- equilateral and so on..but i'd like to know the logical way to do it...

if A,B,C are the vertices of a triangle,
and if cosBcosC + sinAsinBsinC = 1

is the triangle
a) equilateral
b) isosceles
c) right angled isosceles
d) right angled but not isosceles
e) none of the above.
 
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  • #2
Right, I'm assuming you know sin^2 +cos^2 = 1.

Well you can see this is nearly similar, and almost exactly the same if B = C. So there's two angles the same.

So now cos^2 + sinA sin^2 = 1

Which means sinA must be 1.

Therefore A can only take value of 90 degrees - right angle (450...etc useless here)

180 - 90..other two angles combine as total 90 degrees, so they must be 45 each since you've already found they're the same. Which leaves you a right angled isosceles.
 
  • #3


Solving trigonometry triangle questions using cosine and sine functions can seem daunting at first, but with practice and understanding of the concepts, it becomes easier. Here are the steps to solve this particular question:

1. Recall the trigonometric identities for cosine and sine functions. In this case, we have cosBcosC + sinAsinBsinC = 1. This is a variation of the cosine double angle identity, which states that cos(A+B) = cosAcosB - sinAsinB. By rearranging the terms, we can see that cosBcosC + sinAsinBsinC = cos(B+C) = 1.

2. Use the given information to determine the value of the angle B+C. In this case, we know that cos(B+C) = 1, which means that the angle (B+C) is either 0 or 360 degrees. However, since we are dealing with a triangle, the sum of all angles must be 180 degrees. Therefore, we can conclude that angle (B+C) = 180 degrees.

3. Use the triangle angle sum theorem to find the value of the remaining angle, A. The triangle angle sum theorem states that the sum of all angles in a triangle is 180 degrees. Therefore, we can calculate A by subtracting (B+C) from 180 degrees. In this case, A = 180 - (B+C) = 180 - 180 = 0 degrees.

4. Use the values of A, B, and C to determine the type of triangle. Since A = 0 degrees, we can conclude that the triangle is a right-angled triangle. Furthermore, since (B+C) = 180 degrees, we can also conclude that the triangle is not isosceles, as the angles B and C must be different. Therefore, the correct answer is d) right-angled but not isosceles.

In conclusion, when solving trigonometry triangle questions using cosine and sine functions, it is important to remember the trigonometric identities, use the given information to determine the values of the angles, and apply the relevant theorems to determine the type of triangle. With practice, you will become more comfortable and confident in solving these types of questions.
 

1. How do I identify which trigonometric function to use?

To determine which trigonometric function to use in solving a triangle, you need to first identify the known and unknown sides and angles of the triangle. If you know two sides and one angle, you can use the sine function. If you know two angles and one side, you can use the cosine function. If you know one angle and its opposite side, you can use the tangent function.

2. What is the process for solving a trigonometric triangle using cosine and sine functions?

The process for solving a trigonometric triangle using cosine and sine functions involves using the given information to set up a trigonometric equation and then using the inverse function to solve for the unknown side or angle. For example, if you know the measures of two angles and one side of a triangle, you can use the cosine function to find the length of the other side by taking the inverse cosine of the ratio of the adjacent side to the hypotenuse.

3. What is the difference between sine and cosine functions?

The sine and cosine functions are both trigonometric functions, but they are used to solve different types of triangles. The sine function is used to find the length of a side when given an angle and its opposite side, while the cosine function is used to find the length of a side when given two angles and one side. Additionally, the sine function is the ratio of the opposite side to the hypotenuse, while the cosine function is the ratio of the adjacent side to the hypotenuse.

4. How do I use the inverse function to solve a trigonometric triangle?

The inverse function, also known as the inverse trigonometric function, is used to solve for the measure of an angle or the length of a side in a right triangle. To use the inverse function, you need to know the ratio of the sides in relation to the given angle. For example, if you know the ratio of the adjacent side to the hypotenuse, you can use the inverse cosine function to find the measure of the adjacent angle.

5. What are some common mistakes to avoid when solving a trigonometric triangle using cosine and sine functions?

One common mistake when solving a trigonometric triangle is forgetting to convert angles from degrees to radians when using a calculator. Another mistake is using the wrong inverse function for the given ratio of sides. It is also important to pay attention to the signs of the ratios, as this can affect the quadrant in which the angle is located. Finally, be sure to check your calculations and answers to ensure they make sense in the context of the problem.

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