A surveyor applies a trigonometric identity to combine two bearing angles. Given angle A (A) = 45.0000 deg; angle B (B) = 28.0000 deg, determine the cos(A-B) (R).

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \cos A \cos B + \sin A \sin B$$

3. Step 3

   $$List the givens: angle A (A) = 45.0000 deg, angle B (B) = 28.0000 deg$$

4. Step 4 — Substitute the given values:

   $$R = \cos 45.0000 \cos 28.0000 + \sin 45.0000 \sin 28.0000$$

5. Step 5 — Evaluate:

   $$R = 0.9563$$

6. Step 6 — Check: returning R = 0.9563 to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.9126 — kept a factor of two that cancels in the correct rearrangement.
• 0.4782 — dropped that same factor in the other direction.
• 1.0519 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

An analyst uses a trig identity to simplify a phase-difference calculation. Given angle B (B) = 23.0000 deg; cos(A-B) (R) = 0.8710, determine the angle A (A) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except A is given, so isolate A symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for A:

   $$A = B + \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle B (B) = 23.0000 deg, \cos (A-B) (R) = 0.8710$$

4. Step 4 — Substitute the given values:

   $$A = 23.0000 + \cos^{-1}(0.8710)$$

5. Step 5 — Evaluate:

   $$A = 52.4249\ \text{deg}$$

6. Step 6 — Check: returning A = 52.4249 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 104.8 — kept a factor of two that cancels in the correct rearrangement.
• 26.2125 — dropped that same factor in the other direction.
• 57.6674 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A student verifies a trigonometric identity by substituting numeric angles. Given angle A (A) = 49.0000 deg; cos(A-B) (R) = 0.6510, determine the angle B (B) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except B is given, so isolate B symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for B:

   $$B = A - \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle A (A) = 49.0000 deg, \cos (A-B) (R) = 0.6510$$

4. Step 4 — Substitute the given values:

   $$B = 49.0000 - \cos^{-1}(0.6510)$$

5. Step 5 — Evaluate:

   $$B = -0.3830\ \text{deg}$$

6. Step 6 — Check: returning B = -0.3830 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -0.7659 — kept a factor of two that cancels in the correct rearrangement.
• -0.1915 — dropped that same factor in the other direction.
• -0.4213 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A surveyor applies a trigonometric identity to combine two bearing angles. Given angle A (A) = 44.0000 deg; angle B (B) = 33.0000 deg, determine the cos(A-B) (R).

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \cos A \cos B + \sin A \sin B$$

3. Step 3

   $$List the givens: angle A (A) = 44.0000 deg, angle B (B) = 33.0000 deg$$

4. Step 4 — Substitute the given values:

   $$R = \cos 44.0000 \cos 33.0000 + \sin 44.0000 \sin 33.0000$$

5. Step 5 — Evaluate:

   $$R = 0.9816$$

6. Step 6 — Check: returning R = 0.9816 to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.9633 — kept a factor of two that cancels in the correct rearrangement.
• 0.4908 — dropped that same factor in the other direction.
• 1.0798 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

An analyst uses a trig identity to simplify a phase-difference calculation. Given angle B (B) = 29.0000 deg; cos(A-B) (R) = 0.8550, determine the angle A (A) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except A is given, so isolate A symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for A:

   $$A = B + \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle B (B) = 29.0000 deg, \cos (A-B) (R) = 0.8550$$

4. Step 4 — Substitute the given values:

   $$A = 29.0000 + \cos^{-1}(0.8550)$$

5. Step 5 — Evaluate:

   $$A = 60.2403\ \text{deg}$$

6. Step 6 — Check: returning A = 60.2403 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 120.5 — kept a factor of two that cancels in the correct rearrangement.
• 30.1201 — dropped that same factor in the other direction.
• 66.2643 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A student verifies a trigonometric identity by substituting numeric angles. Given angle A (A) = 64.0000 deg; cos(A-B) (R) = 0.2910, determine the angle B (B) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except B is given, so isolate B symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for B:

   $$B = A - \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle A (A) = 64.0000 deg, \cos (A-B) (R) = 0.2910$$

4. Step 4 — Substitute the given values:

   $$B = 64.0000 - \cos^{-1}(0.2910)$$

5. Step 5 — Evaluate:

   $$B = -9.0822\ \text{deg}$$

6. Step 6 — Check: returning B = -9.0822 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -18.1643 — kept a factor of two that cancels in the correct rearrangement.
• -4.5411 — dropped that same factor in the other direction.
• -9.9904 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A surveyor applies a trigonometric identity to combine two bearing angles. Given angle A (A) = 53.0000 deg; angle B (B) = 25.0000 deg, determine the cos(A-B) (R).

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \cos A \cos B + \sin A \sin B$$

3. Step 3

   $$List the givens: angle A (A) = 53.0000 deg, angle B (B) = 25.0000 deg$$

4. Step 4 — Substitute the given values:

   $$R = \cos 53.0000 \cos 25.0000 + \sin 53.0000 \sin 25.0000$$

5. Step 5 — Evaluate:

   $$R = 0.8829$$

6. Step 6 — Check: returning R = 0.8829 to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.7659 — kept a factor of two that cancels in the correct rearrangement.
• 0.4415 — dropped that same factor in the other direction.
• 0.9712 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

An analyst uses a trig identity to simplify a phase-difference calculation. Given angle B (B) = 15.0000 deg; cos(A-B) (R) = 0.2390, determine the angle A (A) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except A is given, so isolate A symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for A:

   $$A = B + \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle B (B) = 15.0000 deg, \cos (A-B) (R) = 0.2390$$

4. Step 4 — Substitute the given values:

   $$A = 15.0000 + \cos^{-1}(0.2390)$$

5. Step 5 — Evaluate:

   $$A = 91.1725\ \text{deg}$$

6. Step 6 — Check: returning A = 91.1725 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 182.3 — kept a factor of two that cancels in the correct rearrangement.
• 45.5862 — dropped that same factor in the other direction.
• 100.3 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A student verifies a trigonometric identity by substituting numeric angles. Given angle A (A) = 47.0000 deg; cos(A-B) (R) = 0.9680, determine the angle B (B) in deg.

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except B is given, so isolate B symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for B:

   $$B = A - \cos^{-1}(R)$$

3. Step 3

   $$List the givens: angle A (A) = 47.0000 deg, \cos (A-B) (R) = 0.9680$$

4. Step 4 — Substitute the given values:

   $$B = 47.0000 - \cos^{-1}(0.9680)$$

5. Step 5 — Evaluate:

   $$B = 32.4663\ \text{deg}$$

6. Step 6 — Check: returning B = 32.4663 deg to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 64.9325 — kept a factor of two that cancels in the correct rearrangement.
• 16.2331 — dropped that same factor in the other direction.
• 35.7129 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities

A surveyor applies a trigonometric identity to combine two bearing angles. Given angle A (A) = 75.0000 deg; angle B (B) = 27.0000 deg, determine the cos(A-B) (R).

Start with the thinking

• The governing relation printed in this handbook section is Trigonometric identities — angle difference.
• Everything except R is given, so isolate R symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The trigonometric identity for the cosine of an angle difference relates A, B, and their combined cosine value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

2. Step 2 — Rearrange symbolically for R:

   $$R = \cos A \cos B + \sin A \sin B$$

3. Step 3

   $$List the givens: angle A (A) = 75.0000 deg, angle B (B) = 27.0000 deg$$

4. Step 4 — Substitute the given values:

   $$R = \cos 75.0000 \cos 27.0000 + \sin 75.0000 \sin 27.0000$$

5. Step 5 — Evaluate:

   $$R = 0.6691$$

6. Step 6 — Check: returning R = 0.6691 to

   $$\cos(A-B) = \cos A \cos B + \sin A \sin B$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.3383 — kept a factor of two that cancels in the correct rearrangement.
• 0.3346 — dropped that same factor in the other direction.
• 0.7360 — rounded an intermediate value before the final step.

Reference: FE Handbook — Trigonometric Identities