Architects use trigonometry every day. The three-dimensional nature of architecture requires her to understand how a building works with trigonometric functions. She constantly uses sines, cosines and tangents to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles. Although many structural aspects can be solved using algebra and the Pythagorean theorem, trigonometry is a faster and easier method to find horizontal and vertical components. Because many architectural elements are not vertical or horizontal, the diagonal directionality of a force, slope or ray of light is modelled as the hypotenuse of a right triangle. Use the basic and inverse trigonometric functions to find the horizontal and vertical vectors of this hypotenuse.

- Skill level:
- Moderately Easy

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### Things you need

- Protractor
- Calculator with trigonometric functions
- Pencil
- Paper

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## Instructions

- 1
Calculate the angle of a structural element or truss member with a protractor. Find the direction of the loads on the structure; these are either known or calculated from building code requirements given by local municipalities.

Break the direction of the load into horizontal and vertical components. For the horizontal component, multiply the cosine of the member angle by the total load. For the vertical component, multiply the sine of the member angle by the total load.

Double-check your trigonometry by dividing your vertical component by your horizontal component and taking the inverse tangent of your quotient. The angle should be equal to the member angle.

- 2
Calculate the angle of the roof or ground slope.

Find the elevation change of a specified horizontal distance by multiplying the tangent of the slope angle by the horizontal distance.

Find the horizontal distance of a given elevation change by dividing the elevation change by the tangent of the slope angle.

Double-check your calculations by taking the inverse tangent of the elevation change divided by the horizontal distance -- if the calculated angle is equal to the slope angle, the calculations are correct.

- 3
Calculate the angle of the light coming from the sun or other light source using a protractor.

Find the depth required for an awning or other shading device by dividing the window or aperture height by the tangent of the light angle.

Double-check the angle by finding the inverse tangent of the height divided by the depth. The calculated angle should be equal to the light angle.

- 4
Calculate the angle of the light coming from the sun or other light source using a protractor.

Find the height of an object by multiplying the object's shadow length by the tangent of the light angle.

Double-check the angle by finding the inverse tangent of the calculated height divided by the shadow length. If the calculated angle is equal to the light angle, then the height is correct.

#### Tips and warnings

- Explore the trigonometric relationships of the adjacent, opposite and hypotenuse sides of a right triangle to find other uses of trigonometry in architecture.