Angle of Elevation worksheet

1. Angle of elevation: 28.18 °

1. Divide the rise by the distance, then use inverse tangent (tan⁻¹) to turn that ratio back into the angle of elevation

   $\theta$$=\text{tan⁻¹}\left(\frac{30}{56}\right)$$=28.17859\text{°}$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{56}^2+{30}^2}$$=63.529521$

2. Horizontal distance: 43.18

1. The tangent of the angle of elevation is rise ÷ distance, so divide the rise by the tangent to get the distance

   $\mathrm{distance}$$=\frac{22}{\text{tan}27\text{°}}$$=\frac{22}{0.50952545}$$=43.177431$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{43.177431}^2+{22}^2}$$=48.459164$

3. Horizontal distance: 41.57

1. The tangent of the angle of elevation is rise ÷ distance, so divide the rise by the tangent to get the distance

   $\mathrm{distance}$$=\frac{24}{\text{tan}30\text{°}}$$=\frac{24}{0.57735027}$$=41.569219$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{41.569219}^2+{24}^2}$$=48$

4. Height of the object: 14.73

1. Call the unknown rise h and the distance from the nearer spot x. Each sighting gives its own right triangle, so write x two ways

   $\text{x = h ÷ tan β, x + 17 = h ÷ tan α}$

2. Subtract one from the other so x drops out, then solve for the height h

   $h$$=\frac{17\times 0.53170943\times 1.37638192}{1.37638192-0.53170943}$$=14.729021$

3. Now find how far the nearer spot is from the base: the rise divided by tan β

   $x$$=\frac{14.729021}{\text{tan}54\text{°}}$$=10.70126$

4. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{10.70126}^2+{14.729021}^2}$$=18.206072$

5. Angle of depression: 52.59 °

1. Divide the rise by the distance, then use inverse tangent (tan⁻¹) to turn that ratio back into the angle of depression

   $\theta$$=\text{tan⁻¹}\left(\frac{34}{26}\right)$$=52.594643\text{°}$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{26}^2+{34}^2}$$=42.801869$

6. Horizontal distance: 35.24

1. The tangent of the angle of elevation is rise ÷ distance, so divide the rise by the tangent to get the distance

   $\mathrm{distance}$$=\frac{42}{\text{tan}50\text{°}}$$=\frac{42}{1.19175359}$$=35.242185$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{35.242185}^2+{42}^2}$$=54.827106$

7. Height above the target: 32.69

1. Your line of sight, the flat ground and the upright side make a right triangle. So the tangent of the angle of depression links the rise (how far up) to the distance along the ground

   $\text{tan}36\text{°}$$=\frac{\mathrm{rise}}{\mathrm{distance}}$

2. Multiply the distance by the tangent of the angle to get the rise

   $\mathrm{rise}$$=45\times \text{tan}36\text{°}$$=45\times 0.72654253$$=32.694414$

3. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{45}^2+{32.694414}^2}$$=55.623059$

8. Height above the target: 18.89

1. Your line of sight, the flat ground and the upright side make a right triangle. So the tangent of the angle of depression links the rise (how far up) to the distance along the ground

   $\text{tan}34\text{°}$$=\frac{\mathrm{rise}}{\mathrm{distance}}$

2. Multiply the distance by the tangent of the angle to get the rise

   $\mathrm{rise}$$=28\times \text{tan}34\text{°}$$=28\times 0.67450852$$=18.886238$

3. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{28}^2+{18.886238}^2}$$=33.774103$

9. Height above the target: 23.75

1. Your line of sight, the flat ground and the upright side make a right triangle. So the tangent of the angle of depression links the rise (how far up) to the distance along the ground

   $\text{tan}32\text{°}$$=\frac{\mathrm{rise}}{\mathrm{distance}}$

2. Multiply the distance by the tangent of the angle to get the rise

   $\mathrm{rise}$$=38\times \text{tan}32\text{°}$$=38\times 0.62486935$$=23.745035$

3. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{38}^2+{23.745035}^2}$$=44.808779$

10. Angle of elevation: 49.4 °

1. Divide the rise by the distance, then use inverse tangent (tan⁻¹) to turn that ratio back into the angle of elevation

   $\theta$$=\text{tan⁻¹}\left(\frac{28}{24}\right)$$=49.398705\text{°}$

2. Find the straight-line distance to the top with the Pythagorean theorem

   $\mathrm{sight}$$=\sqrt{{24}^2+{28}^2}$$=36.878178$