The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 3 kN, β = 30°, and γ = 75°, determine the magnitudes of its three components.

#### Solution:

### Let us first substitute the values given in the question into our diagram.

### To calculate the value of \alpha we have to remember the following formula.

### \text{cos}^2\alpha +\text{cos}^2\beta +\text{cos}^2\gamma =1

### Let’s substitute the values we know. Thus, we have:

### \text{cos}^2\alpha +\text{cos}^{2}(30^0)+\text{cos}^{2}(75^0) =1

##### Now we will solve for \alpha by isolating for \alpha:

### \text{cos}^2\alpha=1-\text{cos}^{2}(30^0)-\text{cos}^{2}(75^0)

### \text{cos}^2\alpha=0.183

##### (Take the square root of both sides)

### \text{cos}\,\alpha=0.4278

### \alpha=\text{cos}^{-1}(0.4278)

### \alpha=64.67^0

### To find the magnitude of the components, we simply substitute the angle values like so: