# How Can Trigonometry Calculate Distances in Surveying?

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• OldWorldBlues
In summary, we discussed using trigonometric functions to calculate the height and distance of an object, specifically using the sine and tangent functions. We also learned about the mnemonic device SOH-CAH-TOA to remember the relationships between trig functions and their corresponding sides. Finally, we clarified that θ represents the angle between the adjacent and hypotenuse sides in these formulas.
OldWorldBlues
Hi there! I haven't yet taken a trigonometry course (I'm in High-school), but I have an amateur interest in surveying. Recently I began thinking about how I could calculate the height of a point relative to me, or the distance of the object from me. Naturally, I immediately thought of the Pythagorean Theorem. However, the formula's need for 2 known lengths of a triangle proved unwieldy for my purposes. I did some research, and came across a formula from Clark University:
sinθ = length of opposite / length of adjacent
Doing some algebra, I got:
(sinθ)length of adjacent = length of opposite
Where θ is the acute angle directly adjacent to the triangle's right angle. I worked out a few problems on paper, which seemed to fit together well, but like I said: I'm no expert. Is there anything I need to know about trig functions (special rules, etc.)? Thanks in advance for any help you guys can give me :)

## \sin(\theta)=##length of opposite/hypotenuse . ## \tan(\theta)=## length of opposite/length of adjacent.

## \sin(\theta)=##length of opposite/hypotenuse . ## \tan(\theta)=## length of opposite/length of adjacent.
Oh, darn! Thanks for the correction. I think I meant to write tan() but got mixed up.

There's a simple mnemonic device that's helpful to learn the relationships: SOH-CAH-TOA
It represents these relationships:
##\sin(\theta) = \frac {\text{opposite}}{\text{hypotenuse}}##

If you have these memorized, the other three trig functions are pretty straightforward.
##\csc(\theta) = \frac 1 {\sin(\theta)}##
##\sec(\theta) = \frac 1 {\cos(\theta)}##
##\cot(\theta) = \frac 1 {\tan(\theta)}##

opus
Mark44 said:
There's a simple mnemonic device that's helpful to learn the relationships: SOH-CAH-TOA
It represents these relationships:
##\sin(\theta) = \frac {\text{opposite}}{\text{hypotenuse}}##

If you have these memorized, the other three trig functions are pretty straightforward.
##\csc(\theta) = \frac 1 {\sin(\theta)}##
##\sec(\theta) = \frac 1 {\cos(\theta)}##
##\cot(\theta) = \frac 1 {\tan(\theta)}##
Thanks, that's pretty useful :)
Is there a specific place that θ HAS to be, or is it just the angle between the adjacent & hypotenuse?

Peter Stravinski said:
Thanks, that's pretty useful :)
Is there a specific place that θ HAS to be, or is it just the angle between the adjacent & hypotenuse?
Yes, the formulas I wrote assume that θ is the angle between the adjacent & hypotenuse.

## 1. How do you use trigonometry to find distance?

To find distance using trigonometry, you need to use the formula d = r * θ, where d is the distance, r is the radius, and θ is the angle. You also need to know the values of two of these variables to solve for the third.

## 2. What is the difference between using sine, cosine, and tangent to find distance?

Sine, cosine, and tangent are all trigonometric functions that can be used to find distance. Sine is used when you know the opposite and hypotenuse sides of a right triangle, cosine is used when you know the adjacent and hypotenuse sides, and tangent is used when you know the opposite and adjacent sides. The choice of function depends on the given information and the angle being measured.

## 3. Can trigonometry be used to find distance in any shape?

Trigonometry can only be used to find distance in right triangles, where one angle is 90 degrees. If the shape is not a right triangle, you will need to use other mathematical methods such as the Pythagorean theorem to find the distance.

## 4. How do you use trigonometry to find distance in real-life situations?

In real-life situations, you can use trigonometry to find distance by first identifying the right triangle formed by the known and unknown sides. Then, you can use the trigonometric functions and the given information to solve for the distance. This is commonly used in navigation, architecture, and engineering.

## 5. What are some common mistakes when using trigonometry to find distance?

One common mistake is using the wrong trigonometric function for the given information, which can result in an incorrect answer. Another mistake is forgetting to convert angles to the correct units, such as degrees or radians. It's also important to pay attention to the order of operations when using a calculator to avoid errors.

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